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In Notes 0, we introduced the notion of a measure space , which includes as a special case the notion of a probability space. By selecting one such probability space as a sample space, one obtains a model for random events and random variables, with random events being modeled by measurable sets in , and random variables taking values in a measurable space being modeled by measurable functions . We then defined some basic operations on these random events and variables:
- Given events , we defined the conjunction , the disjunction , and the complement . For countable families of events, we similarly defined and . We also defined the empty event and the sure event , and what it meant for two events to be equal.
- Given random variables in ranges respectively, and a measurable function , we defined the random variable in range . (As the special case of this, every deterministic element of was also a random variable taking values in .) Given a relation , we similarly defined the event . Conversely, given an event , we defined the indicator random variable . Finally, we defined what it meant for two random variables to be equal.
- Given an event , we defined its probability .
These operations obey various axioms; for instance, the boolean operations on events obey the axioms of a Boolean algebra, and the probabilility function obeys the Kolmogorov axioms. However, we will not focus on the axiomatic approach to probability theory here, instead basing the foundations of probability theory on the sample space models as discussed in Notes 0. (But see this previous post for a treatment of one such axiomatic approach.)
It turns out that almost all of the other operations on random events and variables we need can be constructed in terms of the above basic operations. In particular, this allows one to safely extend the sample space in probability theory whenever needed, provided one uses an extension that respects the above basic operations. We gave a simple example of such an extension in the previous notes, but now we give a more formal definition:
Definition 1 Suppose that we are using a probability space as the model for a collection of events and random variables. An extension of this probability space is a probability space , together with a measurable map (sometimes called the factor map) which is probability-preserving in the sense that
for all . (Caution: this does not imply that for all – why not?)
An event which is modeled by a measurable subset in the sample space , will be modeled by the measurable set in the extended sample space . Similarly, a random variable taking values in some range that is modeled by a measurable function in , will be modeled instead by the measurable function in . We also allow the extension to model additional events and random variables that were not modeled by the original sample space (indeed, this is one of the main reasons why we perform extensions in probability in the first place).
Thus, for instance, the sample space in Example 3 of the previous post is an extension of the sample space in that example, with the factor map given by the first coordinate projection . One can verify that all of the basic operations on events and random variables listed above are unaffected by the above extension (with one caveat, see remark below). For instance, the conjunction of two events can be defined via the original model by the formula
or via the extension via the formula
The two definitions are consistent with each other, thanks to the obvious set-theoretic identity
Similarly, the assumption (1) is precisely what is needed to ensure that the probability of an event remains unchanged when one replaces a sample space model with an extension. We leave the verification of preservation of the other basic operations described above under extension as exercises to the reader.
Remark 2 There is one minor exception to this general rule if we do not impose the additional requirement that the factor map is surjective. Namely, for non-surjective , it can become possible that two events are unequal in the original sample space model, but become equal in the extension (and similarly for random variables), although the converse never happens (events that are equal in the original sample space always remain equal in the extension). For instance, let be the discrete probability space with and , and let be the discrete probability space with , and non-surjective factor map defined by . Then the event modeled by in is distinct from the empty event when viewed in , but becomes equal to that event when viewed in . Thus we see that extending the sample space by a non-surjective factor map can identify previously distinct events together (though of course, being probability preserving, this can only happen if those two events were already almost surely equal anyway). This turns out to be fairly harmless though; while it is nice to know if two given events are equal, or if they differ by a non-null event, it is almost never useful to know that two events are unequal if they are already almost surely equal. Alternatively, one can add the additional requirement of surjectivity in the definition of an extension, which is also a fairly harmless constraint to impose (this is what I chose to do in this previous set of notes).
Roughly speaking, one can define probability theory as the study of those properties of random events and random variables that are model-independent in the sense that they are preserved by extensions. For instance, the cardinality of the model of an event is not a concept within the scope of probability theory, as it is not preserved by extensions: continuing Example 3 from Notes 0, the event that a die roll is even is modeled by a set of cardinality in the original sample space model , but by a set of cardinality in the extension. Thus it does not make sense in the context of probability theory to refer to the “cardinality of an event “.
On the other hand, the supremum of a collection of random variables in the extended real line is a valid probabilistic concept. This can be seen by manually verifying that this operation is preserved under extension of the sample space, but one can also see this by defining the supremum in terms of existing basic operations. Indeed, note from Exercise 24 of Notes 0 that a random variable in the extended real line is completely specified by the threshold events for ; in particular, two such random variables are equal if and only if the events and are surely equal for all . From the identity
we thus see that one can completely specify in terms of using only the basic operations provided in the above list (and in particular using the countable conjunction .) Of course, the same considerations hold if one replaces supremum, by infimum, limit superior, limit inferior, or (if it exists) the limit.
In this set of notes, we will define some further important operations on scalar random variables, in particular the expectation of these variables. In the sample space models, expectation corresponds to the notion of integration on a measure space. As we will need to use both expectation and integration in this course, we will thus begin by quickly reviewing the basics of integration on a measure space, although we will then translate the key results of this theory into probabilistic language.
As the finer details of the Lebesgue integral construction are not the core focus of this probability course, some of the details of this construction will be left to exercises. See also Chapter 1 of Durrett, or these previous blog notes, for a more detailed treatment.
Starting this week, I will be teaching an introductory graduate course (Math 275A) on probability theory here at UCLA. While I find myself using probabilistic methods routinely nowadays in my research (for instance, the probabilistic concept of Shannon entropy played a crucial role in my recent paper on the Chowla and Elliott conjectures, and random multiplicative functions similarly played a central role in the paper on the Erdos discrepancy problem), this will actually be the first time I will be teaching a course on probability itself (although I did give a course on random matrix theory some years ago that presumed familiarity with graduate-level probability theory). As such, I will be relying primarily on an existing textbook, in this case Durrett’s Probability: Theory and Examples. I still need to prepare lecture notes, though, and so I thought I would continue my practice of putting my notes online, although in this particular case they will be less detailed or complete than with other courses, as they will mostly be focusing on those topics that are not already comprehensively covered in the text of Durrett. Below the fold are my first such set of notes, concerning the classical measure-theoretic foundations of probability. (I wrote on these foundations also in this previous blog post, but in that post I already assumed that the reader was familiar with measure theory and basic probability, whereas in this course not every student will have a strong background in these areas.)
Note: as this set of notes is primarily concerned with foundational issues, it will contain a large number of pedantic (and nearly trivial) formalities and philosophical points. We dwell on these technicalities in this set of notes primarily so that they are out of the way in later notes, when we work with the actual mathematics of probability, rather than on the supporting foundations of that mathematics. In particular, the excessively formal and philosophical language in this set of notes will not be replicated in later notes.
In analytic number theory, there is a well known analogy between the prime factorisation of a large integer, and the cycle decomposition of a large permutation; this analogy is central to the topic of “anatomy of the integers”, as discussed for instance in this survey article of Granville. Consider for instance the following two parallel lists of facts (stated somewhat informally). Firstly, some facts about the prime factorisation of large integers:
- Every positive integer has a prime factorisation
into (not necessarily distinct) primes , which is unique up to rearrangement. Taking logarithms, we obtain a partition
- (Prime number theorem) A randomly selected integer of size will be prime with probability when is large.
- If is a randomly selected large integer of size , and is a randomly selected prime factor of (with each index being chosen with probability ), then is approximately uniformly distributed between and . (See Proposition 9 of this previous blog post.)
- The set of real numbers arising from the prime factorisation of a large random number converges (away from the origin, and in a suitable weak sense) to the Poisson-Dirichlet process in the limit . (See the previously mentioned blog post for a definition of the Poisson-Dirichlet process, and a proof of this claim.)
Now for the facts about the cycle decomposition of large permutations:
- Every permutation has a cycle decomposition
into disjoint cycles , which is unique up to rearrangement, and where we count each fixed point of as a cycle of length . If is the length of the cycle , we obtain a partition
- (Prime number theorem for permutations) A randomly selected permutation of will be an -cycle with probability exactly . (This was noted in this previous blog post.)
- If is a random permutation in , and is a randomly selected cycle of (with each being selected with probability ), then is exactly uniformly distributed on . (See Proposition 8 of this blog post.)
- The set of real numbers arising from the cycle decomposition of a random permutation converges (in a suitable sense) to the Poisson-Dirichlet process in the limit . (Again, see this previous blog post for details.)
See this previous blog post (or the aforementioned article of Granville, or the Notices article of Arratia, Barbour, and Tavaré) for further exploration of the analogy between prime factorisation of integers and cycle decomposition of permutations.
There is however something unsatisfying about the analogy, in that it is not clear why there should be such a kinship between integer prime factorisation and permutation cycle decomposition. It turns out that the situation is clarified if one uses another fundamental analogy in number theory, namely the analogy between integers and polynomials over a finite field , discussed for instance in this previous post; this is the simplest case of the more general function field analogy between number fields and function fields. Just as we restrict attention to positive integers when talking about prime factorisation, it will be reasonable to restrict attention to monic polynomials . We then have another analogous list of facts, proven very similarly to the corresponding list of facts for the integers:
- Every monic polynomial has a factorisation
into irreducible monic polynomials , which is unique up to rearrangement. Taking degrees, we obtain a partition
- (Prime number theorem for polynomials) A randomly selected monic polynomial of degree will be irreducible with probability when is fixed and is large.
- If is a random monic polynomial of degree , and is a random irreducible factor of (with each selected with probability ), then is approximately uniformly distributed in when is fixed and is large.
- The set of real numbers arising from the factorisation of a randomly selected polynomial of degree converges (in a suitable sense) to the Poisson-Dirichlet process when is fixed and is large.
The above list of facts addressed the large limit of the polynomial ring , where the order of the field is held fixed, but the degrees of the polynomials go to infinity. This is the limit that is most closely analogous to the integers . However, there is another interesting asymptotic limit of polynomial rings to consider, namely the large limit where it is now the degree that is held fixed, but the order of the field goes to infinity. Actually to simplify the exposition we will use the slightly more restrictive limit where the characteristic of the field goes to infinity (again keeping the degree fixed), although all of the results proven below for the large limit turn out to be true as well in the large limit.
The large (or large ) limit is technically a different limit than the large limit, but in practice the asymptotic statistics of the two limits often agree quite closely. For instance, here is the prime number theorem in the large limit:
Proof: There are monic polynomials of degree . If is irreducible, then the zeroes of are distinct and lie in the finite field , but do not lie in any proper subfield of that field. Conversely, every element of that does not lie in a proper subfield is the root of a unique monic polynomial in of degree (the minimal polynomial of ). Since the union of all the proper subfields of has size , the total number of irreducible polynomials of degree is thus , and the claim follows.
Remark 2 The above argument and inclusion-exclusion in fact gives the well known exact formula for the number of irreducible monic polynomials of degree .
Now we can give a precise connection between the cycle distribution of a random permutation, and (the large limit of) the irreducible factorisation of a polynomial, giving a (somewhat indirect, but still connected) link between permutation cycle decomposition and integer factorisation:
Theorem 3 The partition of a random monic polynomial of degree converges in distribution to the partition of a random permutation of length , in the limit where is fixed and the characteristic goes to infinity.
We can quickly prove this theorem as follows. We first need a basic fact:
Lemma 4 (Most polynomials square-free in large limit) A random monic polynomial of degree will be square-free with probability when is fixed and (or ) goes to infinity. In a similar spirit, two randomly selected monic polynomials of degree will be coprime with probability if are fixed and or goes to infinity.
Proof: For any polynomial of degree , the probability that is divisible by is at most . Summing over all polynomials of degree , and using the union bound, we see that the probability that is not squarefree is at most , giving the first claim. For the second, observe from the first claim (and the fact that has only a bounded number of factors) that is squarefree with probability , giving the claim.
Now we can prove the theorem. Elementary combinatorics tells us that the probability of a random permutation consisting of cycles of length for , where are nonnegative integers with , is precisely
since there are ways to write a given tuple of cycles in cycle notation in nondecreasing order of length, and ways to select the labels for the cycle notation. On the other hand, by Theorem 1 (and using Lemma 4 to isolate the small number of cases involving repeated factors) the number of monic polynomials of degree that are the product of irreducible polynomials of degree is
which simplifies to
and the claim follows.
This was a fairly short calculation, but it still doesn’t quite explain why there is such a link between the cycle decomposition of permutations and the factorisation of a polynomial. One immediate thought might be to try to link the multiplication structure of permutations in with the multiplication structure of polynomials; however, these structures are too dissimilar to set up a convincing analogy. For instance, the multiplication law on polynomials is abelian and non-invertible, whilst the multiplication law on is (extremely) non-abelian but invertible. Also, the multiplication of a degree and a degree polynomial is a degree polynomial, whereas the group multiplication law on permutations does not take a permutation in and a permutation in and return a permutation in .
I recently found (after some discussions with Ben Green) what I feel to be a satisfying conceptual (as opposed to computational) explanation of this link, which I will place below the fold.
I’ve just uploaded to the arXiv my paper “Inverse theorems for sets and measures of polynomial growth“. This paper was motivated by two related questions. The first question was to obtain a qualitatively precise description of the sets of polynomial growth that arise in Gromov’s theorem, in much the same way that Freiman’s theorem (and its generalisations) provide a qualitatively precise description of sets of small doubling. The other question was to obtain a non-abelian analogue of inverse Littlewood-Offord theory.
Let me discuss the former question first. Gromov’s theorem tells us that if a finite subset of a group exhibits polynomial growth in the sense that grows polynomially in , then the group generated by is virtually nilpotent (the converse direction also true, and is relatively easy to establish). This theorem has been strengthened a number of times over the years. For instance, a few years ago, I proved with Shalom that the condition that grew polynomially in could be replaced by for a single , as long as was sufficiently large depending on (in fact we gave a fairly explicit quantitative bound on how large needed to be). A little more recently, with Breuillard and Green, the condition was weakened to , that is to say it sufficed to have polynomial relative growth at a finite scale. In fact, the latter paper gave more information on in this case, roughly speaking it showed (at least in the case when was a symmetric neighbourhood of the identity) that was “commensurate” with a very structured object known as a coset nilprogression. This can then be used to establish further control on . For instance, it was recently shown by Breuillard and Tointon (again in the symmetric case) that if for a single that was sufficiently large depending on , then all the for have a doubling constant bounded by a bound depending only on , thus for all .
In this paper we are able to refine this analysis a bit further; under the same hypotheses, we can show an estimate of the form
for all and some piecewise linear, continuous, non-decreasing function with , where the error is bounded by a constant depending only on , and where has at most pieces, each of which has a slope that is a natural number of size . To put it another way, the function for behaves (up to multiplicative constants) like a piecewise polynomial function, where the degree of the function and number of pieces is bounded by a constant depending on .
One could ask whether the function has any convexity or concavity properties. It turns out that it can exhibit either convex or concave behaviour (or a combination of both). For instance, if is contained in a large finite group, then will eventually plateau to a constant, exhibiting concave behaviour. On the other hand, in nilpotent groups one can see convex behaviour; for instance, in the Heisenberg group , if one sets to be a set of matrices of the form for some large (abusing the notation somewhat), then grows cubically for but then grows quartically for .
To prove this proposition, it turns out (after using a somewhat difficult inverse theorem proven previously by Breuillard, Green, and myself) that one has to analyse the volume growth of nilprogressions . In the “infinitely proper” case where there are no unexpected relations between the generators of the nilprogression, one can lift everything to a simply connected Lie group (where one can take logarithms and exploit the Baker-Campbell-Hausdorff formula heavily), eventually describing with fair accuracy by a certain convex polytope with vertices depending polynomially on , which implies that depends polynomially on up to constants. If one is not in the “infinitely proper” case, then at some point the nilprogression develops a “collision”, but then one can use this collision to show (after some work) that the dimension of the “Lie model” of has dropped by at least one from the dimension of (the notion of a Lie model being developed in the previously mentioned paper of Breuillard, Greenm, and myself), so that this sort of collision can only occur a bounded number of times, with essentially polynomial volume growth behaviour between these collisions.
The arguments also give a precise description of the location of a set for which grows polynomially in . In the symmetric case, what ends up happening is that becomes commensurate to a “coset nilprogression” of bounded rank and nilpotency class, whilst is “virtually” contained in a scaled down version of that nilprogression. What “virtually” means is a little complicated; roughly speaking, it means that there is a set of bounded cardinality such that for all . Conversely, if is virtually contained in , then is commensurate to (and more generally, is commensurate to for any natural number ), giving quite a (qualitatively) precise description of in terms of coset nilprogressions.
The main tool used to prove these results is the structure theorem for approximate groups established by Breuillard, Green, and myself, which roughly speaking asserts that approximate groups are always commensurate with coset nilprogressions. A key additional trick is a pigeonholing argument of Sanders, which in this context is the assertion that if is comparable to , then there is an between and such that is very close in size to (up to a relative error of ). It is this fact, together with the comparability of to a coset nilprogression , that allows us (after some combinatorial argument) to virtually place inside .
Similar arguments apply when discussing iterated convolutions of (symmetric) probability measures on a (discrete) group , rather than combinatorial powers of a finite set. Here, the analogue of volume is given by the negative power of the norm of (thought of as a non-negative function on of total mass 1). One can also work with other norms here than , but this norm has some minor technical conveniences (and other measures of the “spread” of end up being more or less equivalent for our purposes). There is an analogous structure theorem that asserts that if spreads at most polynomially in , then is “commensurate” with the uniform probability distribution on a coset progression , and itself is largely concentrated near . The factor of here is the familiar scaling factor in random walks that arises for instance in the central limit theorem. The proof of (the precise version of) this statement proceeds similarly to the combinatorial case, using pigeonholing to locate a scale where has almost the same norm as .
A special case of this theory occurs when is the uniform probability measure on elements of and their inverses. The probability measure is then the distribution of a random product , where each is equal to one of or its inverse , selected at random with drawn uniformly from with replacement. This is very close to the Littlewood-Offord situation of random products where each is equal to or selected independently at random (thus is now fixed to equal rather than being randomly drawn from . In the case when is abelian, it turns out that a little bit of Fourier analysis shows that these two random walks have “comparable” distributions in a certain sense. As a consequence, the results in this paper can be used to recover an essentially optimal abelian inverse Littlewood-Offord theorem of Nguyen and Vu. In the nonabelian case, the only Littlewood-Offord theorem I am aware of is a recent result of Tiep and Vu for matrix groups, but in this case I do not know how to relate the above two random walks to each other, and so we can only obtain an analogue of the Tiep-Vu results for the symmetrised random walk instead of the ordered random walk .
Hoi Nguyen, Van Vu, and myself have just uploaded to the arXiv our paper “Random matrices: tail bounds for gaps between eigenvalues“. This is a followup paper to my recent paper with Van in which we showed that random matrices of Wigner type (such as the adjacency matrix of an Erdös-Renyi graph) asymptotically almost surely had simple spectrum. In the current paper, we push the method further to show that the eigenvalues are not only distinct, but are (with high probability) separated from each other by some negative power of . This follows the now standard technique of replacing any appearance of discrete Littlewood-Offord theory (a key ingredient in our previous paper) with its continuous analogue (inverse theorems for small ball probability). For general Wigner-type matrices (in which the matrix entries are not normalised to have mean zero), we can use the inverse Littlewood-Offord theorem of Nguyen and Vu to obtain (under mild conditions on ) a result of the form
for any and , if is sufficiently large depending on (in a linear fashion), and is sufficiently large depending on . The point here is that can be made arbitrarily large, and also that no continuity or smoothness hypothesis is made on the distribution of the entries. (In the continuous case, one can use the machinery of Wegner estimates to obtain results of this type, as was done in a paper of Erdös, Schlein, and Yau.)
In the mean zero case, it becomes more efficient to use an inverse Littlewood-Offord theorem of Rudelson and Vershynin to obtain (with the normalisation that the entries of have unit variance, so that the eigenvalues of are with high probability), giving the bound
for (one also has good results of this type for smaller values of ). This is only optimal in the regime ; we expect to establish some eigenvalue repulsion, improving the RHS to for real matrices and for complex matrices, but this appears to be a more difficult task (possibly requiring some quadratic inverse Littlewood-Offord theory, rather than just linear inverse Littlewood-Offord theory). However, we can get some repulsion if one works with larger gaps, getting a result roughly of the form
for (one also has good results of this type for smaller values of ). This is only optimal in the regime ; we expect to establish some eigenvalue repulsion, improving the RHS to for real matrices and for complex matrices, but this appears to be a more difficult task (possibly requiring some quadratic inverse Littlewood-Offord theory, rather than just linear inverse Littlewood-Offord theory). However, we can get some repulsion if one works with larger gaps, getting a result roughly of the form
for any fixed and some absolute constant (which we can asymptotically make to be for large , though it ought to be as large as ), by using a higher-dimensional version of the Rudelson-Vershynin inverse Littlewood-Offord theorem.
In the case of Erdös-Renyi graphs, we don’t have mean zero and the Rudelson-Vershynin Littlewood-Offord theorem isn’t quite applicable, but by working carefully through the approach based on the Nguyen-Vu theorem we can almost recover (1), except for a loss of on the RHS.
As a sample applications of the eigenvalue separation results, we can now obtain some information about eigenvectors; for instance, we can show that the components of the eigenvectors all have magnitude at least for some with high probability. (Eigenvectors become much more stable, and able to be studied in isolation, once their associated eigenvalue is well separated from the other eigenvalues; see this previous blog post for more discussion.)
We now move away from the world of multiplicative prime number theory covered in Notes 1 and Notes 2, and enter the wider, and complementary, world of non-multiplicative prime number theory, in which one studies statistics related to non-multiplicative patterns, such as twins . This creates a major jump in difficulty; for instance, even the most basic multiplicative result about the primes, namely Euclid’s theorem that there are infinitely many of them, remains unproven for twin primes. Of course, the situation is even worse for stronger results, such as Euler’s theorem, Dirichlet’s theorem, or the prime number theorem. Finally, even many multiplicative questions about the primes remain open. The most famous of these is the Riemann hypothesis, which gives the asymptotic (see Proposition 24 from Notes 2). But even if one assumes the Riemann hypothesis, the precise distribution of the error term in the above asymptotic (or in related asymptotics, such as for the sum that measures the distribution of primes in short intervals) is not entirely clear.
Despite this, we do have a number of extremely convincing and well supported models for the primes (and related objects) that let us predict what the answer to many prime number theory questions (both multiplicative and non-multiplicative) should be, particularly in asymptotic regimes where one can work with aggregate statistics about the primes, rather than with a small number of individual primes. These models are based on taking some statistical distribution related to the primes (e.g. the primality properties of a randomly selected -tuple), and replacing that distribution by a model distribution that is easy to compute with (e.g. a distribution with strong joint independence properties). One can then predict the asymptotic value of various (normalised) statistics about the primes by replacing the relevant statistical distributions of the primes with their simplified models. In this non-rigorous setting, many difficult conjectures on the primes reduce to relatively simple calculations; for instance, all four of the (still unsolved) Landau problems may now be justified in the affirmative by one or more of these models. Indeed, the models are so effective at this task that analytic number theory is in the curious position of being able to confidently predict the answer to a large proportion of the open problems in the subject, whilst not possessing a clear way forward to rigorously confirm these answers!
As it turns out, the models for primes that have turned out to be the most accurate in practice are random models, which involve (either explicitly or implicitly) one or more random variables. This is despite the prime numbers being obviously deterministic in nature; no coins are flipped or dice rolled to create the set of primes. The point is that while the primes have a lot of obvious multiplicative structure (for instance, the product of two primes is never another prime), they do not appear to exhibit much discernible non-multiplicative structure asymptotically, in the sense that they rarely exhibit statistical anomalies in the asymptotic limit that cannot be easily explained in terms of the multiplicative properties of the primes. As such, when considering non-multiplicative statistics of the primes, the primes appear to behave pseudorandomly, and can thus be modeled with reasonable accuracy by a random model. And even for multiplicative problems, which are in principle controlled by the zeroes of the Riemann zeta function, one can obtain good predictions by positing various pseudorandomness properties of these zeroes, so that the distribution of these zeroes can be modeled by a random model.
Of course, one cannot expect perfect accuracy when replicating a deterministic set such as the primes by a probabilistic model of that set, and each of the heuristic models we discuss below have some limitations to the range of statistics about the primes that they can expect to track with reasonable accuracy. For instance, many of the models about the primes do not fully take into account the multiplicative structure of primes, such as the connection with a zeta function with a meromorphic continuation to the entire complex plane; at the opposite extreme, we have the GUE hypothesis which appears to accurately model the zeta function, but does not capture such basic properties of the primes as the fact that the primes are all natural numbers. Nevertheless, each of the models described below, when deployed within their sphere of reasonable application, has (possibly after some fine-tuning) given predictions that are in remarkable agreement with numerical computation and with known rigorous theoretical results, as well as with other models in overlapping spheres of application; they are also broadly compatible with the general heuristic (discussed in this previous post) that in the absence of any exploitable structure, asymptotic statistics should default to the most “uniform”, “pseudorandom”, or “independent” distribution allowable.
As hinted at above, we do not have a single unified model for the prime numbers (other than the primes themselves, of course), but instead have an overlapping family of useful models that each appear to accurately describe some, but not all, aspects of the prime numbers. In this set of notes, we will discuss four such models:
- The Cramér random model and its refinements, which model the set of prime numbers by a random set.
- The Möbius pseudorandomness principle, which predicts that the Möbius function does not correlate with any genuinely different arithmetic sequence of reasonable “complexity”.
- The equidistribution of residues principle, which predicts that the residue classes of a large number modulo a small or medium-sized prime behave as if they are independently and uniformly distributed as varies.
- The GUE hypothesis, which asserts that the zeroes of the Riemann zeta function are distributed (at microscopic and mesoscopic scales) like the zeroes of a GUE random matrix, and which generalises the pair correlation conjecture regarding pairs of such zeroes.
This is not an exhaustive list of models for the primes and related objects; for instance, there is also the model in which the major arc contribution in the Hardy-Littlewood circle method is predicted to always dominate, and with regards to various finite groups of number-theoretic importance, such as the class groups discussed in Supplement 1, there are also heuristics of Cohen-Lenstra type. Historically, the first heuristic discussion of the primes along these lines was by Sylvester, who worked informally with a model somewhat related to the equidistribution of residues principle. However, we will not discuss any of these models here.
A word of warning: the discussion of the above four models will inevitably be largely informal, and “fuzzy” in nature. While one can certainly make precise formalisations of at least some aspects of these models, one should not be inflexibly wedded to a specific such formalisation as being “the” correct way to pin down the model rigorously. (To quote the statistician George Box: “all models are wrong, but some are useful”.) Indeed, we will see some examples below the fold in which some finer structure in the prime numbers leads to a correction term being added to a “naive” implementation of one of the above models to make it more accurate, and it is perfectly conceivable that some further such fine-tuning will be applied to one or more of these models in the future. These sorts of mathematical models are in some ways closer in nature to the scientific theories used to model the physical world, than they are to the axiomatic theories one is accustomed to in rigorous mathematics, and one should approach the discussion below accordingly. In particular, and in contrast to the other notes in this course, the material here is not directly used for proving further theorems, which is why we have marked it as “optional” material. Nevertheless, the heuristics and models here are still used indirectly for such purposes, for instance by
- giving a clearer indication of what results one expects to be true, thus guiding one to fruitful conjectures;
- providing a quick way to scan for possible errors in a mathematical claim (e.g. by finding that the main term is off from what a model predicts, or an error term is too small);
- gauging the relative strength of various assertions (e.g. classifying some results as “unsurprising”, others as “potential breakthroughs” or “powerful new estimates”, others as “unexpected new phenomena”, and yet others as “way too good to be true”); or
- setting up heuristic barriers (such as the parity barrier) that one has to resolve before resolving certain key problems (e.g. the twin prime conjecture).
See also my previous essay on the distinction between “rigorous” and “post-rigorous” mathematics, or Thurston’s essay discussing, among other things, the “definition-theorem-proof” model of mathematics and its limitations.
Remark 1 The material in this set of notes presumes some prior exposure to probability theory. See for instance this previous post for a quick review of the relevant concepts.
Van Vu and I have just uploaded to the arXiv our paper “Random matrices have simple spectrum“. Recall that an Hermitian matrix is said to have simple eigenvalues if all of its eigenvalues are distinct. This is a very typical property of matrices to have: for instance, as discussed in this previous post, in the space of all Hermitian matrices, the space of matrices without all eigenvalues simple has codimension three, and for real symmetric cases this space has codimension two. In particular, given any random matrix ensemble of Hermitian or real symmetric matrices with an absolutely continuous distribution, we conclude that random matrices drawn from this ensemble will almost surely have simple eigenvalues.
For discrete random matrix ensembles, though, the above argument breaks down, even though general universality heuristics predict that the statistics of discrete ensembles should behave similarly to those of continuous ensembles. A model case here is the adjacency matrix of an Erdös-Rényi graph – a graph on vertices in which any pair of vertices has an independent probability of being in the graph. For the purposes of this paper one should view as fixed, e.g. , while is an asymptotic parameter going to infinity. In this context, our main result is the following (answering a question of Babai):
Our argument works for more general Wigner-type matrix ensembles, but for sake of illustration we will stick with the Erdös-Renyi case. Previous work on local universality for such matrix models (e.g. the work of Erdos, Knowles, Yau, and Yin) was able to show that any individual eigenvalue gap did not vanish with probability (in fact for some absolute constant ), but because there are different gaps that one has to simultaneously ensure to be non-zero, this did not give Theorem 1 as one is forced to apply the union bound.
Our argument in fact gives simplicity of the spectrum with probability for any fixed ; in a subsequent paper we also show that it gives a quantitative lower bound on the eigenvalue gaps (analogous to how many results on the singularity probability of random matrices can be upgraded to a bound on the least singular value).
The basic idea of argument can be sketched as follows. Suppose that has a repeated eigenvalue . We split
for a random minor and a random sign vector ; crucially, and are independent. If has a repeated eigenvalue , then by the Cauchy interlacing law, also has an eigenvalue . We now write down the eigenvector equation for at :
Extracting the top coefficients, we obtain
If we let be the -eigenvector of , then by taking inner products with we conclude that
we typically expect to be non-zero, in which case we arrive at
In other words, in order for to have a repeated eigenvalue, the top right column of has to be orthogonal to an eigenvector of the minor . Note that and are going to be independent (once we specify which eigenvector of to take as ). On the other hand, thanks to inverse Littlewood-Offord theory (specifically, we use an inverse Littlewood-Offord theorem of Nguyen and Vu), we know that the vector is unlikely to be orthogonal to any given vector independent of , unless the coefficients of are extremely special (specifically, that most of them lie in a generalised arithmetic progression). The main remaining difficulty is then to show that eigenvectors of a random matrix are typically not of this special form, and this relies on a conditioning argument originally used by Komlós to bound the singularity probability of a random sign matrix. (Basically, if an eigenvector has this special form, then one can use a fraction of the rows and columns of the random matrix to determine the eigenvector completely, while still preserving enough randomness in the remaining portion of the matrix so that this vector will in fact not be an eigenvector with high probability.)
The 2014 Fields medallists have just been announced as (in alphabetical order of surname) Artur Avila, Manjul Bhargava, Martin Hairer, and Maryam Mirzakhani (see also these nice video profiles for the winners, which is a new initiative of the IMU and the Simons foundation). This time four years ago, I wrote a blog post discussing one result from each of the 2010 medallists; I thought I would try to repeat the exercise here, although the work of the medallists this time around is a little bit further away from my own direct area of expertise than last time, and so my discussion will unfortunately be a bit superficial (and possibly not completely accurate) in places. As before, I am picking these results based on my own idiosyncratic tastes, and they should not be viewed as necessarily being the “best” work of these medallists. (See also the press releases for Avila, Bhargava, Hairer, and Mirzakhani.)
Artur Avila works in dynamical systems and in the study of Schrödinger operators. The work of Avila that I am most familiar with is his solution with Svetlana Jitormiskaya of the ten martini problem of Kac, the solution to which (according to Barry Simon) he offered ten martinis for, hence the name. The problem involves perhaps the simplest example of a Schrödinger operator with non-trivial spectral properties, namely the almost Mathieu operator defined for parameters and by a discrete one-dimensional Schrödinger operator with cosine potential:
This is a bounded self-adjoint operator and thus has a spectrum that is a compact subset of the real line; it arises in a number of physical contexts, most notably in the theory of the integer quantum Hall effect, though I will not discuss these applications here. Remarkably, the structure of this spectrum depends crucially on the Diophantine properties of the frequency . For instance, if is a rational number, then the operator is periodic with period , and then basic (discrete) Floquet theory tells us that the spectrum is simply the union of (possibly touching) intervals. But for irrational (in which case the spectrum is independent of the phase ), the situation is much more fractal in nature, for instance in the critical case the spectrum (as a function of ) gives rise to the Hofstadter butterfly. The “ten martini problem” asserts that for every irrational and every choice of coupling constant , the spectrum is homeomorphic to a Cantor set. Prior to the work of Avila and Jitormiskaya, there were a number of partial results on this problem, notably the result of Puig establishing Cantor spectrum for a full measure set of parameters , as well as results requiring a perturbative hypothesis, such as being very small or very large. The result was also already known for being either very close to rational (i.e. a Liouville number) or very far from rational (a Diophantine number), although the analyses for these two cases failed to meet in the middle, leaving some cases untreated. The argument uses a wide variety of existing techniques, both perturbative and non-perturbative, to attack this problem, as well as an amusing argument by contradiction: they assume (in certain regimes) that the spectrum fails to be a Cantor set, and use this hypothesis to obtain additional Lipschitz control on the spectrum (as a function of the frequency ), which they can then use (after much effort) to improve existing arguments and conclude that the spectrum was in fact Cantor after all!
Manjul Bhargava produces amazingly beautiful mathematics, though most of it is outside of my own area of expertise. One part of his work that touches on an area of my own interest (namely, random matrix theory) is his ongoing work with many co-authors on modeling (both conjecturally and rigorously) the statistics of various key number-theoretic features of elliptic curves (such as their rank, their Selmer group, or their Tate-Shafarevich groups). For instance, with Kane, Lenstra, Poonen, and Rains, Manjul has proposed a very general random matrix model that predicts all of these statistics (for instance, predicting that the -component of the Tate-Shafarevich group is distributed like the cokernel of a certain random -adic matrix, very much in the spirit of the Cohen-Lenstra heuristics discussed in this previous post). But what is even more impressive is that Manjul and his coauthors have been able to verify several non-trivial fragments of this model (e.g. showing that certain moments have the predicted asymptotics), giving for the first time non-trivial upper and lower bounds for various statistics, for instance obtaining lower bounds on how often an elliptic curve has rank or rank , leading most recently (in combination with existing work of Gross-Zagier and of Kolyvagin, among others) to his amazing result with Skinner and Zhang that at least of all elliptic curves over (ordered by height) obey the Birch and Swinnerton-Dyer conjecture. Previously it was not even known that a positive proportion of curves obeyed the conjecture. This is still a fair ways from resolving the conjecture fully (in particular, the situation with the presumably small number of curves of rank and higher is still very poorly understood, and the theory of Gross-Zagier and Kolyvagin that this work relies on, which was initially only available for , has only been extended to totally real number fields thus far, by the work of Zhang), but it certainly does provide hope that the conjecture could be within reach in a statistical sense at least.
Martin Hairer works in at the interface between probability and partial differential equations, and in particular in the theory of stochastic differential equations (SDEs). The result of his that is closest to my own interests is his remarkable demonstration with Jonathan Mattingly of unique invariant measure for the two-dimensional stochastically forced Navier-Stokes equation
on the two-torus , where is a Gaussian field that forces a fixed set of frequencies. It is expected that for any reasonable choice of initial data, the solution to this equation should asymptotically be distributed according to Kolmogorov’s power law, as discussed in this previous post. This is still far from established rigorously (although there are some results in this direction for dyadic models, see e.g. this paper of Cheskidov, Shvydkoy, and Friedlander). However, Hairer and Mattingly were able to show that there was a unique probability distribution to almost every initial data would converge to asymptotically; by the ergodic theorem, this is equivalent to demonstrating the existence and uniqueness of an invariant measure for the flow. Existence can be established using standard methods, but uniqueness is much more difficult. One of the standard routes to uniqueness is to establish a “strong Feller property” that enforces some continuity on the transition operators; among other things, this would mean that two ergodic probability measures with intersecting supports would in fact have a non-trivial common component, contradicting the ergodic theorem (which forces different ergodic measures to be mutually singular). Since all ergodic measures for Navier-Stokes can be seen to contain the origin in their support, this would give uniqueness. Unfortunately, the strong Feller property is unlikely to hold in the infinite-dimensional phase space for Navier-Stokes; but Hairer and Mattingly develop a clean abstract substitute for this property, which they call the asymptotic strong Feller property, which is again a regularity property on the transition operator; this in turn is then demonstrated by a careful application of Malliavin calculus.
Maryam Mirzakhani has mostly focused on the geometry and dynamics of Teichmuller-type moduli spaces, such as the moduli space of Riemann surfaces with a fixed genus and a fixed number of cusps (or with a fixed number of boundaries that are geodesics of a prescribed length). These spaces have an incredibly rich structure, ranging from geometric structure (such as the Kahler geometry given by the Weil-Petersson metric), to dynamical structure (through the action of the mapping class group on this and related spaces), to algebraic structure (viewing these spaces as algebraic varieties), and are thus connected to many other objects of interest in geometry and dynamics. For instance, by developing a new recursive formula for the Weil-Petersson volume of this space, Mirzakhani was able to asymptotically count the number of simple prime geodesics of length up to some threshold in a hyperbolic surface (or more precisely, she obtained asymptotics for the number of such geodesics in a given orbit of the mapping class group); the answer turns out to be polynomial in , in contrast to the much larger class of non-simple prime geodesics, whose asymptotics are exponential in (the “prime number theorem for geodesics”, developed in a classic series of works by Delsart, Huber, Selberg, and Margulis); she also used this formula to establish a new proof of a conjecture of Witten on intersection numbers that was first proven by Kontsevich. More recently, in two lengthy papers with Eskin and with Eskin-Mohammadi, Mirzakhani established rigidity theorems for the action of on such moduli spaces that are close analogues of Ratner’s celebrated rigidity theorems for unipotently generated groups (discussed in this previous blog post). Ratner’s theorems are already notoriously difficult to prove, and rely very much on the polynomial stability properties of unipotent flows; in this even more complicated setting, the unipotent flows are no longer tractable, and Mirzakhani instead uses a recent “exponential drift” method of Benoist and Quint with as a substitute. Ratner’s theorems are incredibly useful for all sorts of problems connected to homogeneous dynamics, and the analogous theorems established by Mirzakhani, Eskin, and Mohammadi have a similarly broad range of applications, for instance in counting periodic billiard trajectories in rational polygons.
In the traditional foundations of probability theory, one selects a probability space , and makes a distinction between deterministic mathematical objects, which do not depend on the sampled state , and stochastic (or random) mathematical objects, which do depend (but in a measurable fashion) on the sampled state . For instance, a deterministic real number would just be an element , whereas a stochastic real number (or real random variable) would be a measurable function , where in this post will always be endowed with the Borel -algebra. (For readers familiar with nonstandard analysis, the adjectives “deterministic” and “stochastic” will be used here in a manner analogous to the uses of the adjectives “standard” and “nonstandard” in nonstandard analysis. The analogy is particularly close when comparing with the “cheap nonstandard analysis” discussed in this previous blog post. We will also use “relative to ” as a synonym for “stochastic”.)
Actually, for our purposes we will adopt the philosophy of identifying stochastic objects that agree almost surely, so if one was to be completely precise, we should define a stochastic real number to be an equivalence class of measurable functions , up to almost sure equivalence. However, we shall often abuse notation and write simply as .
More generally, given any measurable space , we can talk either about deterministic elements , or about stochastic elements of , that is to say equivalence classes of measurable maps up to almost sure equivalence. We will use to denote the set of all stochastic elements of . (For readers familiar with sheaves, it may helpful for the purposes of this post to think of as the space of measurable global sections of the trivial –bundle over .) Of course every deterministic element of can also be viewed as a stochastic element given by (the equivalence class of) the constant function , thus giving an embedding of into . We do not attempt here to give an interpretation of for sets that are not equipped with a -algebra .
Remark 1 In my previous post on the foundations of probability theory, I emphasised the freedom to extend the sample space to a larger sample space whenever one wished to inject additional sources of randomness. This is of course an important freedom to possess (and in the current formalism, is the analogue of the important operation of base change in algebraic geometry), but in this post we will focus on a single fixed sample space , and not consider extensions of this space, so that one only has to consider two types of mathematical objects (deterministic and stochastic), as opposed to having many more such types, one for each potential choice of sample space (with the deterministic objects corresponding to the case when the sample space collapses to a point).
Any (measurable) -ary operation on deterministic mathematical objects then extends to their stochastic counterparts by applying the operation pointwise. For instance, the addition operation on deterministic real numbers extends to an addition operation , by defining the class for to be the equivalence class of the function ; this operation is easily seen to be well-defined. More generally, any measurable -ary deterministic operation between measurable spaces extends to an stochastic operation in the obvious manner.
There is a similar story for -ary relations , although here one has to make a distinction between a deterministic reading of the relation and a stochastic one. Namely, if we are given stochastic objects for , the relation does not necessarily take values in the deterministic Boolean algebra , but only in the stochastic Boolean algebra – thus may be true with some positive probability and also false with some positive probability (with the event that being stochastically true being determined up to null events). Of course, the deterministic Boolean algebra embeds in the stochastic one, so we can talk about a relation being determinstically true or deterministically false, which (due to our identification of stochastic objects that agree almost surely) means that is almost surely true or almost surely false respectively. For instance given two stochastic objects , one can view their equality relation as having a stochastic truth value. This is distinct from the way the equality symbol is used in mathematical logic, which we will now call “equality in the deterministic sense” to reduce confusion. Thus, in the deterministic sense if and only if the stochastic truth value of is equal to , that is to say that for almost all .
Any universal identity for deterministic operations (or universal implication between identities) extends to their stochastic counterparts: for instance, addition is commutative, associative, and cancellative on the space of deterministic reals , and is therefore commutative, associative, and cancellative on stochastic reals as well. However, one has to be more careful when working with mathematical laws that are not expressible as universal identities, or implications between identities. For instance, is an integral domain: if are deterministic reals such that , then one must have or . However, if are stochastic reals such that (in the deterministic sense), then it is no longer necessarily the case that (in the deterministic sense) or that (in the deterministic sense); however, it is still true that “ or ” is true in the deterministic sense if one interprets the boolean operator “or” stochastically, thus “ or ” is true for almost all . Another way to properly obtain a stochastic interpretation of the integral domain property of is to rewrite it as
and then make all sets stochastic to obtain the true statement
thus we have to allow the index for which vanishing occurs to also be stochastic, rather than deterministic. (A technical note: when one proves this statement, one has to select in a measurable fashion; for instance, one can choose to equal when , and otherwise (so that in the “tie-breaking” case when and both vanish, one always selects to equal ).)
Similarly, the law of the excluded middle fails when interpreted deterministically, but remains true when interpreted stochastically: if is a stochastic statement, then it is not necessarily the case that is either deterministically true or deterministically false; however the sentence “ or not-” is still deterministically true if the boolean operator “or” is interpreted stochastically rather than deterministically.
To avoid having to keep pointing out which operations are interpreted stochastically and which ones are interpreted deterministically, we will use the following convention: if we assert that a mathematical sentence involving stochastic objects is true, then (unless otherwise specified) we mean that is deterministically true, assuming that all relations used inside are interpreted stochastically. For instance, if are stochastic reals, when we assert that “Exactly one of , , or is true”, then by default it is understood that the relations , , and the boolean operator “exactly one of” are interpreted stochastically, and the assertion is that the sentence is deterministically true.
In the above discussion, the stochastic objects being considered were elements of a deterministic space , such as the reals . However, it can often be convenient to generalise this situation by allowing the ambient space to also be stochastic. For instance, one might wish to consider a stochastic vector inside a stochastic vector space , or a stochastic edge of a stochastic graph . In order to formally describe this situation within the classical framework of measure theory, one needs to place all the ambient spaces inside a measurable space. This can certainly be done in many contexts (e.g. when considering random graphs on a deterministic set of vertices, or if one is willing to work up to equivalence and place the ambient spaces inside a suitable moduli space), but is not completely natural in other contexts. For instance, if one wishes to consider stochastic vector spaces of potentially unbounded dimension (in particular, potentially larger than any given cardinal that one might specify in advance), then the class of all possible vector spaces is so large that it becomes a proper class rather than a set (even if one works up to equivalence), making it problematic to give this class the structure of a measurable space; furthermore, even once one does so, one needs to take additional care to pin down what it would mean for a random vector lying in a random vector space to depend “measurably” on .
Of course, in any reasonable application one can avoid the set theoretic issues at least by various ad hoc means, for instance by restricting the dimension of all spaces involved to some fixed cardinal such as . However, the measure-theoretic issues can require some additional effort to resolve properly.
In this post I would like to describe a different way to formalise stochastic spaces, and stochastic elements of these spaces, by viewing the spaces as measure-theoretic analogue of a sheaf, but being over the probability space rather than over a topological space; stochastic objects are then sections of such sheaves. Actually, for minor technical reasons it is convenient to work in the slightly more general setting in which the base space is a finite measure space rather than a probability space, thus can take any value in rather than being normalised to equal . This will allow us to easily localise to subevents of without the need for normalisation, even when is a null event (though we caution that the map from deterministic objects ceases to be injective in this latter case). We will however still continue to use probabilistic terminology. despite the lack of normalisation; thus for instance, sets in will be referred to as events, the measure of such a set will be referred to as the probability (which is now permitted to exceed in some cases), and an event whose complement is a null event shall be said to hold almost surely. It is in fact likely that almost all of the theory below extends to base spaces which are -finite rather than finite (for instance, by damping the measure to become finite, without introducing any further null events), although we will not pursue this further generalisation here.
The approach taken in this post is “topos-theoretic” in nature (although we will not use the language of topoi explicitly here), and is well suited to a “pointless” or “point-free” approach to probability theory, in which the role of the stochastic state is suppressed as much as possible; instead, one strives to always adopt a “relative point of view”, with all objects under consideration being viewed as stochastic objects relative to the underlying base space . In this perspective, the stochastic version of a set is as follows.
Definition 1 (Stochastic set) Unless otherwise specified, we assume that we are given a fixed finite measure space (which we refer to as the base space). A stochastic set (relative to ) is a tuple consisting of the following objects:
- A set assigned to each event ; and
- A restriction map from to to each pair of nested events . (Strictly speaking, one should indicate the dependence on in the notation for the restriction map, e.g. using instead of , but we will abuse notation by omitting the dependence.)
We refer to elements of as local stochastic elements of the stochastic set , localised to the event , and elements of as global stochastic elements (or simply elements) of the stochastic set. (In the language of sheaves, one would use “sections” instead of “elements” here, but I prefer to use the latter terminology here, for compatibility with conventional probabilistic notation, where for instance measurable maps from to are referred to as real random variables, rather than sections of the reals.)
Furthermore, we impose the following axioms:
- (Category) The map from to is the identity map, and if are events in , then for all .
- (Null events trivial) If is a null event, then the set is a singleton set. (In particular, is always a singleton set; this is analogous to the convention that for any number .)
- (Countable gluing) Suppose that for each natural number , one has an event and an element such that for all . Then there exists a unique such that for all .
If is an event in , we define the localisation of the stochastic set to to be the stochastic set
relative to . (Note that there is no need to renormalise the measure on , as we are not demanding that our base space have total measure .)
The following fact is useful for actually verifying that a given object indeed has the structure of a stochastic set:
Exercise 1 Show that to verify the countable gluing axiom of a stochastic set, it suffices to do so under the additional hypothesis that the events are disjoint. (Note that this is quite different from the situation with sheaves over a topological space, in which the analogous gluing axiom is often trivial in the disjoint case but has non-trivial content in the overlapping case. This is ultimately because a -algebra is closed under all Boolean operations, whereas a topology is only closed under union and intersection.)
Let us illustrate the concept of a stochastic set with some examples.
Example 1 (Discrete case) A simple case arises when is a discrete space which is at most countable. If we assign a set to each , with a singleton if . One then sets , with the obvious restriction maps, giving rise to a stochastic set . (Thus, a local element of can be viewed as a map on that takes values in for each .) Conversely, it is not difficult to see that any stochastic set over an at most countable discrete probability space is of this form up to isomorphism. In this case, one can think of as a bundle of sets over each point (of positive probability) in the base space . One can extend this bundle interpretation of stochastic sets to reasonably nice sample spaces (such as standard Borel spaces) and similarly reasonable ; however, I would like to avoid this interpretation in the formalism below in order to be able to easily work in settings in which and are very “large” (e.g. not separable in any reasonable sense). Note that we permit some of the to be empty, thus it can be possible for to be empty whilst for some strict subevents of to be non-empty. (This is analogous to how it is possible for a sheaf to have local sections but no global sections.) As such, the space of global elements does not completely determine the stochastic set ; one sometimes needs to localise to an event in order to see the full structure of such a set. Thus it is important to distinguish between a stochastic set and its space of global elements. (As such, it is a slight abuse of the axiom of extensionality to refer to global elements of simply as “elements”, but hopefully this should not cause too much confusion.)
Example 2 (Measurable spaces as stochastic sets) Returning now to a general base space , any (deterministic) measurable space gives rise to a stochastic set , with being defined as in previous discussion as the measurable functions from to modulo almost everywhere equivalence (in particular, a singleton set when is null), with the usual restriction maps. The constraint of measurability on the maps , together with the quotienting by almost sure equivalence, means that is now more complicated than a plain Cartesian product of fibres, but this still serves as a useful first approximation to what is for the purposes of developing intuition. Indeed, the measurability constraint is so weak (as compared for instance to topological or smooth constraints in other contexts, such as sheaves of continuous or smooth sections of bundles) that the intuition of essentially independent fibres is quite an accurate one, at least if one avoids consideration of an uncountable number of objects simultaneously.
Example 3 (Extended Hilbert modules) This example is the one that motivated this post for me. Suppose that one has an extension of the base space , thus we have a measurable factor map such that the pushforward of the measure by is equal to . Then we have a conditional expectation operator , defined as the adjoint of the pullback map . As is well known, the conditional expectation operator also extends to a contraction ; by monotone convergence we may also extend to a map from measurable functions from to the extended non-negative reals , to measurable functions from to . We then define the “extended Hilbert module” to be the space of functions with finite almost everywhere. This is an extended version of the Hilbert module , which is defined similarly except that is required to lie in ; this is a Hilbert module over which is of particular importance in the Furstenberg-Zimmer structure theory of measure-preserving systems. We can then define the stochastic set by setting
with the obvious restriction maps. In the case that are standard Borel spaces, one can disintegrate as an integral of probability measures (supported in the fibre ), in which case this stochastic set can be viewed as having fibres (though if is not discrete, there are still some measurability conditions in on the local and global elements that need to be imposed). However, I am interested in the case when are not standard Borel spaces (in fact, I will take them to be algebraic probability spaces, as defined in this previous post), in which case disintegrations are not available. However, it appears that the stochastic analysis developed in this blog post can serve as a substitute for the tool of disintegration in this context.
We make the remark that if is a stochastic set and are events that are equivalent up to null events, then one can identify with (through their common restriction to , with the restriction maps now being bijections). As such, the notion of a stochastic set does not require the full structure of a concrete probability space ; one could also have defined the notion using only the abstract -algebra consisting of modulo null events as the base space, or equivalently one could define stochastic sets over the algebraic probability spaces defined in this previous post. However, we will stick with the classical formalism of concrete probability spaces here so as to keep the notation reasonably familiar.
As a corollary of the above observation, we see that if the base space has total measure , then all stochastic sets are trivial (they are just points).
Exercise 2 If is a stochastic set, show that there exists an event with the property that for any event , is non-empty if and only if is contained in modulo null events. (In particular, is unique up to null events.) Hint: consider the numbers for ranging over all events with non-empty, and form a maximising sequence for these numbers. Then use all three axioms of a stochastic set.
One can now start take many of the fundamental objects, operations, and results in set theory (and, hence, in most other categories of mathematics) and establish analogues relative to a finite measure space. Implicitly, what we will be doing in the next few paragraphs is endowing the category of stochastic sets with the structure of an elementary topos. However, to keep things reasonably concrete, we will not explicitly emphasise the topos-theoretic formalism here, although it is certainly lurking in the background.
Firstly, we define a stochastic function between two stochastic sets to be a collection of maps for each which form a natural transformation in the sense that for all and nested events . In the case when is discrete and at most countable (and after deleting all null points), a stochastic function is nothing more than a collection of functions for each , with the function then being a direct sum of the factor functions :
Thus (in the discrete, at most countable setting, at least) stochastic functions do not mix together information from different states in a sample space; the value of at depends only on the value of at . The situation is a bit more subtle for continuous probability spaces, due to the identification of stochastic objects that agree almost surely, nevertheness it is still good intuition to think of stochastic functions as essentially being “pointwise” or “local” in nature.
One can now form the stochastic set of functions from to , by setting for any event to be the set of local stochastic functions of the localisations of to ; this is a stochastic set if we use the obvious restriction maps. In the case when is discrete and at most countable, the fibre at a point of positive measure is simply the set of functions from to .
In a similar spirit, we say that one stochastic set is a (stochastic) subset of another , and write , if we have a stochastic inclusion map, thus for all events , with the restriction maps being compatible. We can then define the power set of a stochastic set by setting for any event to be the set of all stochastic subsets of relative to ; it is easy to see that is a stochastic set with the obvious restriction maps (one can also identify with in the obvious fashion). Again, when is discrete and at most countable, the fibre of at a point of positive measure is simply the deterministic power set .
Note that if is a stochastic function and is a stochastic subset of , then the inverse image , defined by setting for any event to be the set of those with , is a stochastic subset of . In particular, given a -ary relation , the inverse image is a stochastic subset of , which by abuse of notation we denote as
In a similar spirit, if is a stochastic subset of and is a stochastic function, we can define the image by setting to be the set of those with ; one easily verifies that this is a stochastic subset of .
Remark 2 One should caution that in the definition of the subset relation , it is important that for all events , not just the global event ; in particular, just because a stochastic set has no global sections, does not mean that it is contained in the stochastic empty set .
Now we discuss Boolean operations on stochastic subsets of a given stochastic set . Given two stochastic subsets of , the stochastic intersection is defined by setting to be the set of that lie in both and :
This is easily verified to again be a stochastic subset of . More generally one may define stochastic countable intersections for any sequence of stochastic subsets of . One could extend this definition to uncountable families if one wished, but I would advise against it, because some of the usual laws of Boolean algebra (e.g. the de Morgan laws) may break down in this setting.
Stochastic unions are a bit more subtle. The set should not be defined to simply be the union of and , as this would not respect the gluing axiom. Instead, we define to be the set of all such that one can cover by measurable subevents such that for ; then may be verified to be a stochastic subset of . Thus for instance is the stochastic union of and . Similarly for countable unions of stochastic subsets of , although for uncountable unions are extremely problematic (they are disliked by both the measure theory and the countable gluing axiom) and will not be defined here. Finally, the stochastic difference set is defined as the set of all in such that for any subevent of of positive probability. One may verify that in the case when is discrete and at most countable, these Boolean operations correspond to the classical Boolean operations applied separately to each fibre of the relevant sets . We also leave as an exercise to the reader to verify the usual laws of Boolean arithmetic, e.g. the de Morgan laws, provided that one works with at most countable unions and intersections.
One can also consider a stochastic finite union in which the number of sets in the union is itself stochastic. More precisely, let be a stochastic set, let be a stochastic natural number, and let be a stochastic function from the stochastic set (defined by setting )) to the stochastic power set . Here we are considering to be a natural number, to allow for unions that are possibly empty, with used for the positive natural numbers. We also write for the stochastic function . Then we can define the stochastic union by setting for an event to be the set of local elements with the property that there exists a covering of by measurable subevents for , such that one has and . One can verify that is a stochastic set (with the obvious restriction maps). Again, in the model case when is discrete and at most countable, the fibre is what one would expect it to be, namely .
The Cartesian product of two stochastic sets may be defined by setting for all events , with the obvious restriction maps; this is easily seen to be another stochastic set. This lets one define the concept of a -ary operation from stochastic sets to another stochastic set , or a -ary relation . In particular, given for , the relation may be deterministically true, deterministically false, or have some other stochastic truth value.
Remark 3 In the degenerate case when is null, stochastic logic becomes a bit weird: all stochastic statements are deterministically true, as are their stochastic negations, since every event in (even the empty set) now holds with full probability. Among other pathologies, the empty set now has a global element over (this is analogous to the notorious convention ), and any two deterministic objects become equal over : .
The following simple observation is crucial to subsequent discussion. If is a sequence taking values in the global elements of a stochastic space , then we may also define global elements for stochastic indices as well, by appealing to the countable gluing axiom to glue together restricted to the set for each deterministic natural number to form . With this definition, the map is a stochastic function from to ; indeed, this creates a one-to-one correspondence between external sequences (maps from to ) and stochastic sequences (stochastic functions from to ). Similarly with replaced by any other at most countable set. This observation will be important in allowing many deterministic arguments involving sequences will be able to be carried over to the stochastic setting.
We now specialise from the extremely broad discipline of set theory to the more focused discipline of real analysis. There are two fundamental axioms that underlie real analysis (and in particular distinguishes it from real algebra). The first is the Archimedean property, which we phrase in the “no infinitesimal” formulation as follows:
Proposition 2 (Archimedean property) Let be such that for all positive natural numbers . Then .
The other is the least upper bound axiom:
- for all .
- For any real , there exists such that .
Furthermore, does not depend on the choice of .
The Archimedean property extends easily to the stochastic setting:
Remark 4 Here, incidentally, is one place in which this stochastic formalism deviates from the nonstandard analysis formalism, as the latter certainly permits the existence of infinitesimal elements. On the other hand, we caution that stochastic real numbers are permitted to be unbounded, so that formulation of Archimedean property is not valid in the stochastic setting.
The proof is easy and is left to the reader. The least upper bound axiom also extends nicely to the stochastic setting, but the proof requires more work (in particular, our argument uses the monotone convergence theorem):
Theorem 5 (Stochastic least upper bound axiom) Let be a stochastic subset of which has a global upper bound , thus for all , and is globally non-empty in the sense that there is at least one global element . Then there exists a unique stochastic real number with the following properties:
- for all .
- For any stochastic real , there exists such that .
Furthermore, does not depend on the choice of .
For future reference, we note that the same result holds with replaced by throughout, since the latter may be embedded in the former, for instance by mapping to and to . In applications, the above theorem serves as a reasonable substitute for the countable axiom of choice, which does not appear to hold in unrestricted generality relative to a measure space; in particular, it can be used to generate various extremising sequences for stochastic functionals on various stochastic function spaces.
Proof: Uniqueness is clear (using the Archimedean property), as well as the independence on , so we turn to existence. By using an order-preserving map from to (e.g. ) we may assume that is a subset of , and that .
We observe that is a lattice: if , then and also lie in . Indeed, may be formed by appealing to the countable gluing axiom to glue (restricted the set ) with (restricted to the set ), and similarly for . (Here we use the fact that relations such as are Borel measurable on .)
Let denote the deterministic quantity
then (by Proposition 3!) is well-defined; here we use the hypothesis that is finite. Thus we may find a sequence of elements of such that
Using the lattice property, we may assume that the are non-decreasing: whenever . If we then define (after choosing measurable representatives of each equivalence class ), then is a stochastic real with .
Using the lattice property, we may assume that the are non-decreasing: whenever . If we then define (after choosing measurable representatives of each equivalence class ), then is a stochastic real with .
If , then , and so
From this and (1) we conclude that
From monotone convergence, we conclude that
and so , as required.
Now let be a stochastic real. After choosing measurable representatives of each relevant equivalence class, we see that for almost every , we can find a natural number with . If we choose to be the first such positive natural number when it exists, and (say) otherwise, then is a stochastic positive natural number and . The claim follows.
Remark 5 One can abstract away the role of the measure here, leaving only the ideal of null sets. The property that the measure is finite is then replaced by the more general property that given any non-empty family of measurable sets, there is an at most countable union of sets in that family that is an upper bound modulo null sets for all elements in that faily.
Using Proposition 4 and Theorem 5, one can then revisit many of the other foundational results of deterministic real analysis, and develop stochastic analogues; we give some examples of this below the fold (focusing on the Heine-Borel theorem and a case of the spectral theorem). As an application of this formalism, we revisit some of the Furstenberg-Zimmer structural theory of measure-preserving systems, particularly that of relatively compact and relatively weakly mixing systems, and interpret them in this framework, basically as stochastic versions of compact and weakly mixing systems (though with the caveat that the shift map is allowed to act non-trivially on the underlying probability space). As this formalism is “point-free”, in that it avoids explicit use of fibres and disintegrations, it will be well suited for generalising this structure theory to settings in which the underlying probability spaces are not standard Borel, and the underlying groups are uncountable; I hope to discuss such generalisations in future blog posts.
Remark 6 Roughly speaking, stochastic real analysis can be viewed as a restricted subset of classical real analysis in which all operations have to be “measurable” with respect to the base space. In particular, indiscriminate application of the axiom of choice is not permitted, and one should largely restrict oneself to performing countable unions and intersections rather than arbitrary unions or intersections. Presumably one can formalise this intuition with a suitable “countable transfer principle”, but I was not able to formulate a clean and general principle of this sort, instead verifying various assertions about stochastic objects by hand rather than by direct transfer from the deterministic setting. However, it would be desirable to have such a principle, since otherwise one is faced with the tedious task of redoing all the foundations of real analysis (or whatever other base theory of mathematics one is going to be working in) in the stochastic setting by carefully repeating all the arguments.
More generally, topos theory is a good formalism for capturing precisely the informal idea of performing mathematics with certain operations, such as the axiom of choice, the law of the excluded middle, or arbitrary unions and intersections, being somehow “prohibited” or otherwise “restricted”.
As laid out in the foundational work of Kolmogorov, a classical probability space (or probability space for short) is a triplet , where is a set, is a -algebra of subsets of , and is a countably additive probability measure on . Given such a space, one can form a number of interesting function spaces, including
- the (real) Hilbert space of square-integrable functions , modulo -almost everywhere equivalence, and with the positive definite inner product ; and
- the unital commutative Banach algebra of essentially bounded functions , modulo -almost everywhere equivalence, with defined as the essential supremum of .
There is also a trace on defined by integration: .
One can form the category of classical probability spaces, by defining a morphism between probability spaces to be a function which is measurable (thus for all ) and measure-preserving (thus for all ).
Let us now abstract the algebraic features of these spaces as follows; for want of a better name, I will refer to this abstraction as an algebraic probability space, and is very similar to the non-commutative probability spaces studied in this previous post, except that these spaces are now commutative (and real).
Definition 1 An algebraic probability space is a pair where
- is a unital commutative real algebra;
- is a homomorphism such that and for all ;
- Every element of is bounded in the sense that . (Technically, this isn’t an algebraic property, but I need it for technical reasons.)
A morphism is a homomorphism which is trace-preserving, in the sense that for all .
For want of a better name, I’ll denote the category of algebraic probability spaces as . One can view this category as the opposite category to that of (a subcategory of) the category of tracial commutative real algebras. One could emphasise this opposite nature by denoting the algebraic probability space as rather than ; another suggestive (but slightly inaccurate) notation, inspired by the language of schemes, would be rather than . However, we will not adopt these conventions here, and refer to algebraic probability spaces just by the pair .
By the previous discussion, we have a covariant functor that takes a classical probability space to its algebraic counterpart , with a morphism of classical probability spaces mapping to a morphism of the corresponding algebraic probability spaces by the formula
for . One easily verifies that this is a functor.
In this post I would like to describe a functor which partially inverts (up to natural isomorphism), that is to say a recipe for starting with an algebraic probability space and producing a classical probability space . This recipe is not new – it is basically the (commutative) Gelfand-Naimark-Segal construction (discussed in this previous post) combined with the Loomis-Sikorski theorem (discussed in this previous post). However, I wanted to put the construction in a single location for sake of reference. I also wanted to make the point that and are not complete inverses; there is a bit of information in the algebraic probability space (e.g. topological information) which is lost when passing back to the classical probability space. In some future posts, I would like to develop some ergodic theory using the algebraic foundations of probability theory rather than the classical foundations; this turns out to be convenient in the ergodic theory arising from nonstandard analysis (such as that described in this previous post), in which the groups involved are uncountable and the underlying spaces are not standard Borel spaces.
Let us describe how to construct the functor , with details postponed to below the fold.
- Starting with an algebraic probability space , form an inner product on by the formula , and also form the spectral radius .
- The inner product is clearly positive semi-definite. Quotienting out the null vectors and taking completions, we arrive at a real Hilbert space , to which the trace may be extended.
- Somewhat less obviously, the spectral radius is well-defined and gives a norm on . Taking limits of sequences in of bounded spectral radius gives us a subspace of that has the structure of a real commutative Banach algebra.
- The idempotents of the Banach algebra may be indexed by elements of an abstract -algebra .
- The Boolean algebra homomorphisms (or equivalently, the real algebra homomorphisms ) may be indexed by elements of a space .
- Let denote the -algebra on generated by the basic sets for every .
- Let be the -ideal of generated by the sets , where is a sequence with .
- One verifies that is isomorphic to . Using this isomorphism, the trace on can be used to construct a countably additive measure on . The classical probability space is then , and the abstract spaces may now be identified with their concrete counterparts , .
- Every algebraic probability space morphism generates a classical probability morphism via the formula
using a pullback operation on the abstract -algebras that can be defined by density.
Remark 1 The classical probability space constructed by the functor has some additional structure; namely is a -Stone space (a Stone space with the property that the closure of any countable union of clopen sets is clopen), is the Baire -algebra (generated by the clopen sets), and the null sets are the meager sets. However, we will not use this additional structure here.
The partial inversion relationship between the functors and is given by the following assertion:
- There is a natural transformation from to the identity functor .
More informally: if one starts with an algebraic probability space and converts it back into a classical probability space , then there is a trace-preserving algebra homomorphism of to , which respects morphisms of the algebraic probability space. While this relationship is far weaker than an equivalence of categories (which would require that and are both natural isomorphisms), it is still good enough to allow many ergodic theory problems formulated using classical probability spaces to be reformulated instead as an equivalent problem in algebraic probability spaces.
Remark 2 The opposite composition is a little odd: it takes an arbitrary probability space and returns a more complicated probability space , with being the space of homomorphisms . while there is “morally” an embedding of into using the evaluation map, this map does not exist in general because points in may well have zero measure. However, if one takes a “pointless” approach and focuses just on the measure algebras , , then these algebras become naturally isomorphic after quotienting out by null sets.
Remark 3 An algebraic probability space captures a bit more structure than a classical probability space, because may be identified with a proper subset of that describes the “regular” functions (or random variables) of the space. For instance, starting with the unit circle (with the usual Haar measure and the usual trace ), any unital subalgebra of that is dense in will generate the same classical probability space on applying the functor , namely one will get the space of homomorphisms from to (with the measure induced from ). Thus for instance could be the continuous functions , the Wiener algebra or the full space , but the classical space will be unable to distinguish these spaces from each other. In particular, the functor loses information (roughly speaking, this functor takes an algebraic probability space and completes it to a von Neumann algebra, but then forgets exactly what algebra was initially used to create this completion). In ergodic theory, this sort of “extra structure” is traditionally encoded in topological terms, by assuming that the underlying probability space has a nice topological structure (e.g. a standard Borel space); however, with the algebraic perspective one has the freedom to have non-topological notions of extra structure, by choosing to be something other than an algebra of continuous functions on a topological space. I hope to discuss one such example of extra structure (coming from the Gowers-Host-Kra theory of uniformity seminorms) in a later blog post (this generalises the example of the Wiener algebra given previously, which is encoding “Fourier structure”).
A small example of how one could use the functors is as follows. Suppose one has a classical probability space with a measure-preserving action of an uncountable group , which is only defined (and an action) up to almost everywhere equivalence; thus for instance for any set and any , and might not be exactly equal, but only equal up to a null set. For similar reasons, an element of the invariant factor might not be exactly invariant with respect to , but instead one only has and equal up to null sets for each . One might like to “clean up” the action of to make it defined everywhere, and a genuine action everywhere, but this is not immediately achievable if is uncountable, since the union of all the null sets where something bad occurs may cease to be a null set. However, by applying the functor , each shift defines a morphism on the associated algebraic probability space (i.e. the Koopman operator), and then applying , we obtain a shift on a new classical probability space which now gives a genuine measure-preserving action of , and which is equivalent to the original action from a measure algebra standpoint. The invariant factor now consists of those sets in which are genuinely -invariant, not just up to null sets. (Basically, the classical probability space contains a Boolean algebra with the property that every measurable set is equivalent up to null sets to precisely one set in , allowing for a canonical “retraction” onto that eliminates all null set issues.)
More indirectly, the functors suggest that one should be able to develop a “pointless” form of ergodic theory, in which the underlying probability spaces are given algebraically rather than classically. I hope to give some more specific examples of this in later posts.