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This is the sixth thread for the Polymath8b project to obtain new bounds for the quantity
either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on can be found at the wiki page (which has recently returned to full functionality, after a partial outage).
The current focus is on improving the upper bound on under the assumption of the generalised Elliott-Halberstam conjecture (GEH) from to , which looks to be the limit of the method (see this previous comment for a semi-rigorous reason as to why is not possible with this method). With the most general Selberg sieve available, the problem reduces to the following three-dimensional variational one:
Problem 1 Does there exist a (not necessarily convex) polytope with quantities , and a non-trivial square-integrable function supported on such that
- when ;
- when ;
- when ;
and such that we have the inequality
(Initially it was assumed that was convex, but we have now realised that this is not necessary.)
An affirmative answer to this question will imply on GEH. We are “within almost two percent” of this claim; we cannot quite reach yet, but have got as far as . However, we have not yet fully optimised in the above problem.
The most promising route so far is to take the symmetric polytope
with symmetric as well, and (we suspect that the optimal will be roughly ). (However, it is certainly worth also taking a look at easier model problems, such as the polytope , which has no vanishing marginal conditions to contend with; more recently we have been looking at the non-convex polytope .) Some further details of this particular case are given below the fold.
There should still be some progress to be made in the other regimes of interest – the unconditional bound on (currently at ), and on any further progress in asymptotic bounds for for larger – but the current focus is certainly on the bound on on GEH, as we seem to be tantalisingly close to an optimal result here.
This is the fifth thread for the Polymath8b project to obtain new bounds for the quantity
either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on can be found at the wiki page (which has recently returned to full functionality, after a partial outage). In particular, the upper bound for has been shaved a little from to , and we have very recently achieved the bound on the generalised Elliott-Halberstam conjecture GEH, formulated as Conjecture 1 of this paper of Bombieri, Friedlander, and Iwaniec. We also have explicit bounds for for , both with and without the assumption of the Elliott-Halberstam conjecture, as well as slightly sharper asymptotics for the upper bound for as .
The basic strategy for bounding still follows the general paradigm first laid out by Goldston, Pintz, Yildirim: given an admissible -tuple , one needs to locate a non-negative sieve weight , supported on an interval for a large , such that the ratio
is asymptotically larger than as ; this will show that . Thus one wants to locate a sieve weight for which one has good lower bounds on the numerator and good upper bounds on the denominator.
One can modify this paradigm slightly, for instance by adding the additional term to the numerator, or by subtracting the term from the numerator (which allows one to reduce the bound to ); however, the numerical impact of these tweaks have proven to be negligible thus far.
Despite a number of experiments with other sieves, we are still relying primarily on the Selberg sieve
where is the divisor sum
with , is the level of distribution ( if relying on Bombieri-Vinogradov, if assuming Elliott-Halberstam, and (in principle) if using Polymath8a technology), and is a smooth, compactly supported function. Most of the progress has come by enlarging the class of cutoff functions one is permitted to use.
The baseline bounds for the numerator and denominator in (1) (as established for instance in this previous post) are as follows. If is supported on the simplex
and we define the mixed partial derivative by
then the denominator in (1) is
and
Similarly, the numerator of (1) is
Thus, if we let be the supremum of the ratio
whenever is supported on and is non-vanishing, then one can prove whenever
We can improve this baseline in a number of ways. Firstly, with regards to the denominator in (1), if one upgrades the Elliott-Halberstam hypothesis to the generalised Elliott-Halberstam hypothesis (currently known for , thanks to Motohashi, but conjectured for ), the asymptotic (2) holds under the more general hypothesis that is supported in a polytope , as long as obeys the inclusion
examples of polytopes obeying this constraint include the modified simplex
the prism
the dilated simplex
and the truncated simplex
See this previous post for a proof of these claims.
With regards to the numerator, the asymptotic (3) is valid whenever, for each , the marginals vanish outside of . This is automatic if is supported on , or on the slightly larger region , but is an additional constraint when is supported on one of the other polytopes mentioned above.
More recently, we have obtained a more flexible version of the above asymptotic: if the marginals vanish outside of for some , then the numerator of (1) has a lower bound of
where
A proof is given here. Putting all this together, we can conclude
Theorem 1 Suppose we can find and a function supported on a polytope obeying (4), not identically zero and with all marginals vanishing outside of , and with
Then implies .
In principle, this very flexible criterion for upper bounding should lead to better bounds than before, and in particular we have now established on GEH.
Another promising direction is to try to improve the analysis at medium (more specifically, in the regime ), which is where we are currently at without EH or GEH through numerical quadratic programming. Right now we are only using and using the baseline analysis, basically for two reasons:
- We do not have good numerical formulae for integrating polynomials on any region more complicated than the simplex in medium dimension.
- The estimates produced by Polymath8a involve a parameter, which introduces additional restrictions on the support of (conservatively, it restricts to where and ; it should be possible to be looser than this (as was done in Polymath8a) but this has not been fully explored yet). This then triggers the previous obstacle of having to integrate on something other than a simplex.
However, these look like solvable problems, and so I would expect that further unconditional improvement for should be possible.
This is the fourth thread for the Polymath8b project to obtain new bounds for the quantity
either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on are:
- (Maynard) Assuming the Elliott-Halberstam conjecture, .
- (Polymath8b, tentative) . Assuming Elliott-Halberstam, .
- (Polymath8b, tentative) . Assuming Elliott-Halberstam, .
- (Polymath8b, tentative) . (Presumably a comparable bound also holds for on Elliott-Halberstam, but this has not been computed.)
- (Polymath8b) for sufficiently large . Assuming Elliott-Halberstam, for sufficiently large .
While the bound on the Elliott-Halberstam conjecture has not improved since the start of the Polymath8b project, there is reason to hope that it will soon fall, hopefully to . This is because we have begun to exploit more fully the fact that when using “multidimensional Selberg-GPY” sieves of the form
with
where , it is not necessary for the smooth function to be supported on the simplex
but can in fact be allowed to range on larger sets. First of all, may instead be supported on the slightly larger polytope
However, it turns out that more is true: given a sufficiently general version of the Elliott-Halberstam conjecture at the given value of , one may work with functions supported on more general domains , so long as the sumset is contained in the non-convex region
and also provided that the restriction
More precisely, if is a smooth function, not identically zero, with the above properties for some , and the ratio
is larger than , then the claim holds (assuming ), and in particular .
I’ll explain why one can do this below the fold. Taking this for granted, we can rewrite this criterion in terms of the mixed derivative , the upshot being that if one can find a smooth function supported on that obeys the vanishing marginal conditions
and
then holds. (To equate these two formulations, it is convenient to assume that is a downset, in the sense that whenever , the entire box lie in , but one can easily enlarge to be a downset without destroying the containment of in the non-convex region (1).) One initially requires to be smooth, but a limiting argument allows one to relax to bounded measurable . (To approximate a rough by a smooth while retaining the required moment conditions, one can first apply a slight dilation and translation so that the marginals of are supported on a slightly smaller version of the simplex , and then convolve by a smooth approximation to the identity to make smooth, while keeping the marginals supported on .)
We are now exploring various choices of to work with, including the prism
and the symmetric region
By suitably subdividing these regions into polytopes, and working with piecewise polynomial functions that are polynomial of a specified degree on each subpolytope, one can phrase the problem of optimising (4) as a quadratic program, which we have managed to work with for . Extending this program to , there is a decent chance that we will be able to obtain on EH.
We have also been able to numerically optimise quite accurately for medium values of (e.g. ), which has led to improved values of without EH. For large , we now also have the asymptotic with explicit error terms (details here) which have allowed us to slightly improve the numerology, and also to get explicit numerology for the first time.
Mertens’ theorems are a set of classical estimates concerning the asymptotic distribution of the prime numbers:
Theorem 1 (Mertens’ theorems) In the asymptotic limit , we have
where is the Euler-Mascheroni constant, defined by requiring that
The third theorem (3) is usually stated in exponentiated form
but in the logarithmic form (3) we see that it is strictly stronger than (2), in view of the asymptotic .
Remarkably, these theorems can be proven without the assistance of the prime number theorem
which was proven about two decades after Mertens’ work. (But one can certainly use versions of the prime number theorem with good error term, together with summation by parts, to obtain good estimates on the various errors in Mertens’ theorems.) Roughly speaking, the reason for this is that Mertens’ theorems only require control on the Riemann zeta function in the neighbourhood of the pole at , whereas (as discussed in this previous post) the prime number theorem requires control on the zeta function on (a neighbourhood of) the line . Specifically, Mertens’ theorem is ultimately deduced from the Euler product formula
valid in the region (which is ultimately a Fourier-Dirichlet transform of the fundamental theorem of arithmetic), and following crude asymptotics:
Proposition 2 (Simple pole) For sufficiently close to with , we have
Proof: For as in the proposition, we have for any natural number and , and hence
Summing in and using the identity , we obtain the first claim. Similarly, we have
and by summing in and using the identity (the derivative of the previous identity) we obtain the claim.
The first two of Mertens’ theorems (1), (2) are relatively easy to prove, and imply the third theorem (3) except with replaced by an unspecified absolute constant. To get the specific constant requires a little bit of additional effort. From (4), one might expect that the appearance of arises from the refinement
that one can obtain to (6). However, it turns out that the connection is not so much with the zeta function, but with the Gamma function, and specifically with the identity (which is of course related to (7) through the functional equation for zeta, but can be proven without any reference to zeta functions). More specifically, we have the following asymptotic for the exponential integral:
Proposition 3 (Exponential integral asymptotics) For sufficiently small , one has
A routine integration by parts shows that this asymptotic is equivalent to the identity
which is the identity mentioned previously.
Proof: We start by using the identity to express the harmonic series as
or on summing the geometric series
Since , we thus have
making the change of variables , this becomes
As , converges pointwise to and is pointwise dominated by . Taking limits as using dominated convergence, we conclude that
or equivalently
The claim then follows by bounding the portion of the integral on the left-hand side.
Below the fold I would like to record how Proposition 2 and Proposition 3 imply Theorem 1; the computations are utterly standard, and can be found in most analytic number theory texts, but I wanted to write them down for my own benefit (I always keep forgetting, in particular, how the third of Mertens’ theorems is proven).
This is the third thread for the Polymath8b project to obtain new bounds for the quantity
either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on are:
- (Maynard) Assuming the Elliott-Halberstam conjecture, .
- (Polymath8b, tentative) . Assuming Elliott-Halberstam, .
- (Polymath8b, tentative) . Assuming Elliott-Halberstam, .
- (Polymath8b) for sufficiently large . Assuming Elliott-Halberstam, for sufficiently large .
Much of the current focus of the Polymath8b project is on the quantity
where ranges over square-integrable functions on the simplex
with being the quadratic forms
and
It was shown by Maynard that one has whenever , where is the narrowest diameter of an admissible -tuple. As discussed in the previous post, we have slight improvements to this implication, but they are currently difficult to implement, due to the need to perform high-dimensional integration. The quantity does seem however to be close to the theoretical limit of what the Selberg sieve method can achieve for implications of this type (at the Bombieri-Vinogradov level of distribution, at least); it seems of interest to explore more general sieves, although we have not yet made much progress in this direction.
The best asymptotic bounds for we have are
which we prove below the fold. The upper bound holds for all ; the lower bound is only valid for sufficiently large , and gives the upper bound on Elliott-Halberstam.
For small , the upper bound is quite competitive, for instance it provides the upper bound in the best values
and
we have for and . The situation is a little less clear for medium values of , for instance we have
and so it is not yet clear whether (which would imply ). See this wiki page for some further upper and lower bounds on .
The best lower bounds are not obtained through the asymptotic analysis, but rather through quadratic programming (extending the original method of Maynard). This has given significant numerical improvements to our best bounds (in particular lowering the bound from to ), but we have not yet been able to combine this method with the other potential improvements (enlarging the simplex, using MPZ distributional estimates, and exploiting upper bounds on two-point correlations) due to the computational difficulty involved.
(This is an extended blog post version of my talk “Ultraproducts as a Bridge Between Discrete and Continuous Analysis” that I gave at the Simons institute for the theory of computing at the workshop “Neo-Classical methods in discrete analysis“. Some of the material here is drawn from previous blog posts, notably “Ultraproducts as a bridge between hard analysis and soft analysis” and “Ultralimit analysis and quantitative algebraic geometry“‘. The text here has substantially more details than the talk; one may wish to skip all of the proofs given here to obtain a closer approximation to the original talk.)
Discrete analysis, of course, is primarily interested in the study of discrete (or “finitary”) mathematical objects: integers, rational numbers (which can be viewed as ratios of integers), finite sets, finite graphs, finite or discrete metric spaces, and so forth. However, many powerful tools in mathematics (e.g. ergodic theory, measure theory, topological group theory, algebraic geometry, spectral theory, etc.) work best when applied to continuous (or “infinitary”) mathematical objects: real or complex numbers, manifolds, algebraic varieties, continuous topological or metric spaces, etc. In order to apply results and ideas from continuous mathematics to discrete settings, there are basically two approaches. One is to directly discretise the arguments used in continuous mathematics, which often requires one to keep careful track of all the bounds on various quantities of interest, particularly with regard to various error terms arising from discretisation which would otherwise have been negligible in the continuous setting. The other is to construct continuous objects as limits of sequences of discrete objects of interest, so that results from continuous mathematics may be applied (often as a “black box”) to the continuous limit, which then can be used to deduce consequences for the original discrete objects which are quantitative (though often ineffectively so). The latter approach is the focus of this current talk.
The following table gives some examples of a discrete theory and its continuous counterpart, together with a limiting procedure that might be used to pass from the former to the latter:
(Discrete) | (Continuous) | (Limit method) |
Ramsey theory | Topological dynamics | Compactness |
Density Ramsey theory | Ergodic theory | Furstenberg correspondence principle |
Graph/hypergraph regularity | Measure theory | Graph limits |
Polynomial regularity | Linear algebra | Ultralimits |
Structural decompositions | Hilbert space geometry | Ultralimits |
Fourier analysis | Spectral theory | Direct and inverse limits |
Quantitative algebraic geometry | Algebraic geometry | Schemes |
Discrete metric spaces | Continuous metric spaces | Gromov-Hausdorff limits |
Approximate group theory | Topological group theory | Model theory |
As the above table illustrates, there are a variety of different ways to form a limiting continuous object. Roughly speaking, one can divide limits into three categories:
- Topological and metric limits. These notions of limits are commonly used by analysts. Here, one starts with a sequence (or perhaps a net) of objects in a common space , which one then endows with the structure of a topological space or a metric space, by defining a notion of distance between two points of the space, or a notion of open neighbourhoods or open sets in the space. Provided that the sequence or net is convergent, this produces a limit object , which remains in the same space, and is “close” to many of the original objects with respect to the given metric or topology.
- Categorical limits. These notions of limits are commonly used by algebraists. Here, one starts with a sequence (or more generally, a diagram) of objects in a category , which are connected to each other by various morphisms. If the ambient category is well-behaved, one can then form the direct limit or the inverse limit of these objects, which is another object in the same category , and is connected to the original objects by various morphisms.
- Logical limits. These notions of limits are commonly used by model theorists. Here, one starts with a sequence of objects or of spaces , each of which is (a component of) a model for given (first-order) mathematical language (e.g. if one is working in the language of groups, might be groups and might be elements of these groups). By using devices such as the ultraproduct construction, or the compactness theorem in logic, one can then create a new object or a new space , which is still a model of the same language (e.g. if the spaces were all groups, then the limiting space will also be a group), and is “close” to the original objects or spaces in the sense that any assertion (in the given language) that is true for the limiting object or space, will also be true for many of the original objects or spaces, and conversely. (For instance, if is an abelian group, then the will also be abelian groups for many .)
The purpose of this talk is to highlight the third type of limit, and specifically the ultraproduct construction, as being a “universal” limiting procedure that can be used to replace most of the limits previously mentioned. Unlike the topological or metric limits, one does not need the original objects to all lie in a common space in order to form an ultralimit ; they are permitted to lie in different spaces ; this is more natural in many discrete contexts, e.g. when considering graphs on vertices in the limit when goes to infinity. Also, no convergence properties on the are required in order for the ultralimit to exist. Similarly, ultraproduct limits differ from categorical limits in that no morphisms between the various spaces involved are required in order to construct the ultraproduct.
With so few requirements on the objects or spaces , the ultraproduct construction is necessarily a very “soft” one. Nevertheless, the construction has two very useful properties which make it particularly useful for the purpose of extracting good continuous limit objects out of a sequence of discrete objects. First of all, there is Łos’s theorem, which roughly speaking asserts that any first-order sentence which is asymptotically obeyed by the , will be exactly obeyed by the limit object ; in particular, one can often take a discrete sequence of “partial counterexamples” to some assertion, and produce a continuous “complete counterexample” that same assertion via an ultraproduct construction; taking the contrapositives, one can often then establish a rigorous equivalence between a quantitative discrete statement and its qualitative continuous counterpart. Secondly, there is the countable saturation property that ultraproducts automatically enjoy, which is a property closely analogous to that of compactness in topological spaces, and can often be used to ensure that the continuous objects produced by ultraproduct methods are “complete” or “compact” in various senses, which is particularly useful in being able to upgrade qualitative (or “pointwise”) bounds to quantitative (or “uniform”) bounds, more or less “for free”, thus reducing significantly the burden of “epsilon management” (although the price one pays for this is that one needs to pay attention to which mathematical objects of study are “standard” and which are “nonstandard”). To achieve this compactness or completeness, one sometimes has to restrict to the “bounded” portion of the ultraproduct, and it is often also convenient to quotient out the “infinitesimal” portion in order to complement these compactness properties with a matching “Hausdorff” property, thus creating familiar examples of continuous spaces, such as locally compact Hausdorff spaces.
Ultraproducts are not the only logical limit in the model theorist’s toolbox, but they are one of the simplest to set up and use, and already suffice for many of the applications of logical limits outside of model theory. In this post, I will set out the basic theory of these ultraproducts, and illustrate how they can be used to pass between discrete and continuous theories in each of the examples listed in the above table.
Apart from the initial “one-time cost” of setting up the ultraproduct machinery, the main loss one incurs when using ultraproduct methods is that it becomes very difficult to extract explicit quantitative bounds from results that are proven by transferring qualitative continuous results to the discrete setting via ultraproducts. However, in many cases (particularly those involving regularity-type lemmas) the bounds are already of tower-exponential type or worse, and there is arguably not much to be lost by abandoning the explicit quantitative bounds altogether.
This is the second thread for the Polymath8b project to obtain new bounds for the quantity
either for small values of (in particular ) or asymptotically as . The previous thread may be found here. The currently best known bounds on are:
- (Maynard) .
- (Polymath8b, tentative) .
- (Polymath8b, tentative) for sufficiently large .
- (Maynard) Assuming the Elliott-Halberstam conjecture, , , and .
Following the strategy of Maynard, the bounds on proceed by combining four ingredients:
- Distribution estimates or for the primes (or related objects);
- Bounds for the minimal diameter of an admissible -tuple;
- Lower bounds for the optimal value to a certain variational problem;
- Sieve-theoretic arguments to convert the previous three ingredients into a bound on .
Accordingly, the most natural routes to improve the bounds on are to improve one or more of the above four ingredients.
Ingredient 1 was studied intensively in Polymath8a. The following results are known or conjectured (see the Polymath8a paper for notation and proofs):
- (Bombieri-Vinogradov) is true for all .
- (Polymath8a) is true for .
- (Polymath8a, tentative) is true for .
- (Elliott-Halberstam conjecture) is true for all .
Ingredient 2 was also studied intensively in Polymath8a, and is more or less a solved problem for the values of of interest (with exact values of for , and quite good upper bounds for for , available at this page). So the main focus currently is on improving Ingredients 3 and 4.
For Ingredient 3, the basic variational problem is to understand the quantity
for bounded measurable functions, not identically zero, on the simplex
with being the quadratic forms
and
Equivalently, one has
where is the positive semi-definite bounded self-adjoint operator
so is the operator norm of . Another interpretation of is that the probability that a rook moving randomly in the unit cube stays in simplex for moves is asymptotically .
We now have a fairly good asymptotic understanding of , with the bounds
holding for sufficiently large . There is however still room to tighten the bounds on for small ; I’ll summarise some of the ideas discussed so far below the fold.
For Ingredient 4, the basic tool is this:
Thus, for instance, it is known that and , and this together with the Bombieri-Vinogradov inequality gives . This result is proven in Maynard’s paper and an alternate proof is also given in the previous blog post.
We have a number of ways to relax the hypotheses of this result, which we also summarise below the fold.
For each natural number , let denote the quantity
where denotes the prime. In other words, is the least quantity such that there are infinitely many intervals of length that contain or more primes. Thus, for instance, the twin prime conjecture is equivalent to the assertion that , and the prime tuples conjecture would imply that is equal to the diameter of the narrowest admissible tuple of cardinality (thus we conjecturally have , , , , , and so forth; see this web page for further continuation of this sequence).
In 2004, Goldston, Pintz, and Yildirim established the bound conditional on the Elliott-Halberstam conjecture, which remains unproven. However, no unconditional finiteness of was obtained (although they famously obtained the non-trivial bound ), and even on the Elliot-Halberstam conjecture no finiteness result on the higher was obtained either (although they were able to show on this conjecture). In the recent breakthrough of Zhang, the unconditional bound was obtained, by establishing a weak partial version of the Elliott-Halberstam conjecture; by refining these methods, the Polymath8 project (which I suppose we could retroactively call the Polymath8a project) then lowered this bound to .
With the very recent preprint of James Maynard, we have the following further substantial improvements:
Theorem 1 (Maynard’s theorem) Unconditionally, we have the following bounds:
- .
- for an absolute constant and any .
If one assumes the Elliott-Halberstam conjecture, we have the following improved bounds:
- .
- .
- for an absolute constant and any .
The final conclusion on Elliott-Halberstam is not explicitly stated in Maynard’s paper, but follows easily from his methods, as I will describe below the fold. (At around the same time as Maynard’s work, I had also begun a similar set of calculations concerning , but was only able to obtain the slightly weaker bound unconditionally.) In the converse direction, the prime tuples conjecture implies that should be comparable to . Granville has also obtained the slightly weaker explicit bound for any by a slight modification of Maynard’s argument.
The arguments of Maynard avoid using the difficult partial results on (weakened forms of) the Elliott-Halberstam conjecture that were established by Zhang and then refined by Polymath8; instead, the main input is the classical Bombieri-Vinogradov theorem, combined with a sieve that is closer in spirit to an older sieve of Goldston and Yildirim, than to the sieve used later by Goldston, Pintz, and Yildirim on which almost all subsequent work is based.
The aim of the Polymath8b project is to obtain improved bounds on , and higher values of , either conditional on the Elliott-Halberstam conjecture or unconditional. The likeliest routes for doing this are by optimising Maynard’s arguments and/or combining them with some of the results from the Polymath8a project. This post is intended to be the first research thread for that purpose. To start the ball rolling, I am going to give below a presentation of Maynard’s results, with some minor technical differences (most significantly, I am using the Goldston-Pintz-Yildirim variant of the Selberg sieve, rather than the traditional “elementary Selberg sieve” that is used by Maynard (and also in the Polymath8 project), although it seems that the numerology obtained by both sieves is essentially the same). An alternate exposition of Maynard’s work has just been completed also by Andrew Granville.
The classical foundations of probability theory (discussed for instance in this previous blog post) is founded on the notion of a probability space – a space (the sample space) equipped with a -algebra (the event space), together with a countably additive probability measure that assigns a real number in the interval to each event.
One can generalise the concept of a probability space to a finitely additive probability space, in which the event space is now only a Boolean algebra rather than a -algebra, and the measure is now only finitely additive instead of countably additive, thus when are disjoint events. By giving up countable additivity, one loses a fair amount of measure and integration theory, and in particular the notion of the expectation of a random variable becomes problematic (unless the random variable takes only finitely many values). Nevertheless, one can still perform a fair amount of probability theory in this weaker setting.
In this post I would like to describe a further weakening of probability theory, which I will call qualitative probability theory, in which one does not assign a precise numerical probability value to each event, but instead merely records whether this probability is zero, one, or something in between. Thus is now a function from to the set , where is a new symbol that replaces all the elements of the open interval . In this setting, one can no longer compute quantitative expressions, such as the mean or variance of a random variable; but one can still talk about whether an event holds almost surely, with positive probability, or with zero probability, and there are still usable notions of independence. (I will refer to classical probability theory as quantitative probability theory, to distinguish it from its qualitative counterpart.)
The main reason I want to introduce this weak notion of probability theory is that it becomes suited to talk about random variables living inside algebraic varieties, even if these varieties are defined over fields other than or . In algebraic geometry one often talks about a “generic” element of a variety defined over a field , which does not lie in any specified variety of lower dimension defined over . Once has positive dimension, such generic elements do not exist as classical, deterministic -points in , since of course any such point lies in the -dimensional subvariety of . There are of course several established ways to deal with this problem. One way (which one might call the “Weil” approach to generic points) is to extend the field to a sufficiently transcendental extension , in order to locate a sufficient number of generic points in . Another approach (which one might dub the “Zariski” approach to generic points) is to work scheme-theoretically, and interpret a generic point in as being associated to the zero ideal in the function ring of . However I want to discuss a third perspective, in which one interprets a generic point not as a deterministic object, but rather as a random variable taking values in , but which lies in any given lower-dimensional subvariety of with probability zero. This interpretation is intuitive, but difficult to implement in classical probability theory (except perhaps when considering varieties over or ) due to the lack of a natural probability measure to place on algebraic varieties; however it works just fine in qualitative probability theory. In particular, the algebraic geometry notion of being “generically true” can now be interpreted probabilistically as an assertion that something is “almost surely true”.
It turns out that just as qualitative random variables may be used to interpret the concept of a generic point, they can also be used to interpret the concept of a type in model theory; the type of a random variable is the set of all predicates that are almost surely obeyed by . In contrast, model theorists often adopt a Weil-type approach to types, in which one works with deterministic representatives of a type, which often do not occur in the original structure of interest, but only in a sufficiently saturated extension of that structure (this is the analogue of working in a sufficiently transcendental extension of the base field). However, it seems that (in some cases at least) one can equivalently view types in terms of (qualitative) random variables on the original structure, avoiding the need to extend that structure. (Instead, one reserves the right to extend the sample space of one’s probability theory whenever necessary, as part of the “probabilistic way of thinking” discussed in this previous blog post.) We illustrate this below the fold with two related theorems that I will interpret through the probabilistic lens: the “group chunk theorem” of Weil (and later developed by Hrushovski), and the “group configuration theorem” of Zilber (and again later developed by Hrushovski). For sake of concreteness we will only consider these theorems in the theory of algebraically closed fields, although the results are quite general and can be applied to many other theories studied in model theory.
One of the basic tools in modern combinatorics is the probabilistic method, introduced by Erdos, in which a deterministic solution to a given problem is shown to exist by constructing a random candidate for a solution, and showing that this candidate solves all the requirements of the problem with positive probability. When the problem requires a real-valued statistic to be suitably large or suitably small, the following trivial observation is often employed:
Proposition 1 (Comparison with mean) Let be a random real-valued variable, whose mean (or first moment) is finite. Then
with positive probability, and
with positive probability.
This proposition is usually applied in conjunction with a computation of the first moment , in which case this version of the probabilistic method becomes an instance of the first moment method. (For comparison with other moment methods, such as the second moment method, exponential moment method, and zeroth moment method, see Chapter 1 of my book with Van Vu. For a general discussion of the probabilistic method, see the book by Alon and Spencer of the same name.)
As a typical example in random matrix theory, if one wanted to understand how small or how large the operator norm of a random matrix could be, one might first try to compute the expected operator norm and then apply Proposition 1; see this previous blog post for examples of this strategy (and related strategies, based on comparing with more tractable expressions such as the moments ). (In this blog post, all matrices are complex-valued.)
Recently, in their proof of the Kadison-Singer conjecture (and also in their earlier paper on Ramanujan graphs), Marcus, Spielman, and Srivastava introduced an striking new variant of the first moment method, suited in particular for controlling the operator norm of a Hermitian positive semi-definite matrix . Such matrices have non-negative real eigenvalues, and so in this case is just the largest eigenvalue of . Traditionally, one tries to control the eigenvalues through averaged statistics such as moments or Stieltjes transforms ; again, see this previous blog post. Here we use as short-hand for , where is the identity matrix. Marcus, Spielman, and Srivastava instead rely on the interpretation of the eigenvalues of as the roots of the characteristic polynomial of , thus
where is the largest real root of a non-zero polynomial . (In our applications, we will only ever apply to polynomials that have at least one real root, but for sake of completeness let us set if has no real roots.)
Prior to the work of Marcus, Spielman, and Srivastava, I think it is safe to say that the conventional wisdom in random matrix theory was that the representation (1) of the operator norm was not particularly useful, due to the highly non-linear nature of both the characteristic polynomial map and the maximum root map . (Although, as pointed out to me by Adam Marcus, some related ideas have occurred in graph theory rather than random matrix theory, for instance in the theory of the matching polynomial of a graph.) For instance, a fact as basic as the triangle inequality is extremely difficult to establish through (1). Nevertheless, it turns out that for certain special types of random matrices (particularly those in which a typical instance of this ensemble has a simple relationship to “adjacent” matrices in this ensemble), the polynomials enjoy an extremely rich structure (in particular, they lie in families of real stable polynomials, and hence enjoy good combinatorial interlacing properties) that can be surprisingly useful. In particular, Marcus, Spielman, and Srivastava established the following nonlinear variant of Proposition 1:
Proposition 2 (Comparison with mean) Let . Let be a random matrix, which is the sum of independent Hermitian rank one matrices , each taking a finite number of values. Then
with positive probability, and
with positive probability.
We prove this proposition below the fold. The hypothesis that each only takes finitely many values is technical and can likely be relaxed substantially, but we will not need to do so here. Despite the superficial similarity with Proposition 1, the proof of Proposition 2 is quite nonlinear; in particular, one needs the interlacing properties of real stable polynomials to proceed. Another key ingredient in the proof is the observation that while the determinant of a matrix generally behaves in a nonlinar fashion on the underlying matrix , it becomes (affine-)linear when one considers rank one perturbations, and so depends in an affine-multilinear fashion on the . More precisely, we have the following deterministic formula, also proven below the fold:
Proposition 3 (Deterministic multilinearisation formula) Let be the sum of deterministic rank one matrices . Then we have
for all , where the mixed characteristic polynomial of any matrices (not necessarily rank one) is given by the formula
Among other things, this formula gives a useful representation of the mean characteristic polynomial :
Corollary 4 (Random multilinearisation formula) Let be the sum of jointly independent rank one matrices . Then we have
Proof: For fixed , the expression is a polynomial combination of the , while the differential operator is a linear combination of differential operators for . As a consequence, we may expand (3) as a linear combination of terms, each of which is a multilinear combination of for some . Taking expectations of both sides of (2) and using the joint independence of the , we obtain the claim.
In view of Proposition 2, we can now hope to control the operator norm of certain special types of random matrices (and specifically, the sum of independent Hermitian positive semi-definite rank one matrices) by first controlling the mean of the random characteristic polynomial . Pursuing this philosophy, Marcus, Spielman, and Srivastava establish the following result, which they then use to prove the Kadison-Singer conjecture:
Theorem 5 (Marcus-Spielman-Srivastava theorem) Let . Let be jointly independent random vectors in , with each taking a finite number of values. Suppose that we have the normalisation
where we are using the convention that is the identity matrix whenever necessary. Suppose also that we have the smallness condition
for some and all . Then one has
Note that the upper bound in (5) must be at least (by taking to be deterministic) and also must be at least (by taking the to always have magnitude at least ). Thus the bound in (5) is asymptotically tight both in the regime and in the regime ; the latter regime will be particularly useful for applications to Kadison-Singer. It should also be noted that if one uses more traditional random matrix theory methods (based on tools such as Proposition 1, as well as more sophisticated variants of these tools, such as the concentration of measure results of Rudelson and Ahlswede-Winter), one obtains a bound of with high probability, which is insufficient for the application to the Kadison-Singer problem; see this article of Tropp. Thus, Theorem 5 obtains a sharper bound, at the cost of trading in “high probability” for “positive probability”.
In the paper of Marcus, Spielman and Srivastava, Theorem 5 is used to deduce a conjecture of Weaver, which was already known to imply the Kadison-Singer conjecture; actually, a slight modification of their argument gives the paving conjecture of Kadison and Singer, from which the original Kadison-Singer conjecture may be readily deduced. We give these implications below the fold. (See also this survey article for some background on the Kadison-Singer problem.)
Let us now summarise how Theorem 5 is proven. In the spirit of semi-definite programming, we rephrase the above theorem in terms of the rank one Hermitian positive semi-definite matrices :
Theorem 6 (Marcus-Spielman-Srivastava theorem again) Let be jointly independent random rank one Hermitian positive semi-definite matrices such that the sum has mean
and such that
for some and all . Then one has
with positive probability.
In view of (1) and Proposition 2, this theorem follows from the following control on the mean characteristic polynomial:
Theorem 7 (Control of mean characteristic polynomial) Let be jointly independent random rank one Hermitian positive semi-definite matrices such that the sum has mean
and such that
for some and all . Then one has
This result is proven using the multilinearisation formula (Corollary 4) and some convexity properties of real stable polynomials; we give the proof below the fold.
Thanks to Adam Marcus, Assaf Naor and Sorin Popa for many useful explanations on various aspects of the Kadison-Singer problem.
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