A function field analogue of Riemann zeta statistics

This is another sequel to a recent post in which I showed the Riemann zeta function can be locally approximated by a polynomial, in the sense that for randomly chosen one has an approximation

where grows slowly with , and is a polynomial of degree . It turns out that in the function field setting there is an exact version of this approximation which captures many of the known features of the Riemann zeta function, namely Dirichlet -functions for a random character of given modulus over a function field. This model was (essentially) studied in a fairly recent paper by Andrade, Miller, Pratt, and Trinh; I am not sure if there is any further literature on this model beyond this paper (though the number field analogue of low-lying zeroes of Dirichlet -functions is certainly well studied). In this model it is possible to set fixed and let go to infinity, thus providing a simple finite-dimensional model problem for problems involving the statistics of zeroes of the zeta function.

In this post I would like to record this analogue precisely. We will need a finite field of some order and a natural number , and set

We will primarily think of as being large and as being either fixed or growing very slowly with , though it is possible to also consider other asymptotic regimes (such as holding fixed and letting go to infinity). Let be the ring of polynomials of one variable with coefficients in , and let be the multiplicative semigroup of monic polynomials in ; one should view and as the function field analogue of the integers and natural numbers respectively. We use the valuation for polynomials (with ); this is the analogue of the usual absolute value on the integers. We select an irreducible polynomial of size (i.e., has degree ). The multiplicative group can be shown to be cyclic of order . A Dirichlet character of modulus is a completely multiplicative function of modulus , that is periodic of period and vanishes on those not coprime to . From Fourier analysis we see that there are exactly Dirichlet characters of modulus . A Dirichlet character is said to be odd if it is not identically one on the group of non-zero constants; there are only non-odd characters (including the principal character), so in the limit most Dirichlet characters are odd. We will work primarily with odd characters in order to be able to ignore the effect of the place at infinity.

Let be an odd Dirichlet character of modulus . The Dirichlet -function is then defined (for of sufficiently large real part, at least) as

Note that for , the set is invariant under shifts whenever ; since this covers a full set of residue classes of , and the odd character has mean zero on this set of residue classes, we conclude that the sum vanishes for . In particular, the -function is entire, and for any real number and complex number , we can write the -function as a polynomial

where and the coefficients are given by the formula

Note that can easily be normalised to zero by the relation

In particular, the dependence on is periodic with period (so by abuse of notation one could also take to be an element of ).

Fourier inversion yields a functional equation for the polynomial :

Proposition 1 (Functional equation) Let be an odd Dirichlet character of modulus , and . There exists a phase (depending on ) such that

for all , or equivalently that

where .

Proof: We can normalise . Let be the finite field . We can write

where denotes the subgroup of consisting of (residue classes of) polynomials of degree less than . Let be a non-trivial character of whose kernel lies in the space (this is easily achieved by pulling back a non-trivial character from the quotient ). We can use the Fourier inversion formula to write

where

From change of variables we see that is a scalar multiple of ; from Plancherel we conclude that

for some phase . We conclude that

The inner sum equals if , and vanishes otherwise, thus

For in , and the contribution of the sum vanishes as is odd. Thus we may restrict to , so that

By the multiplicativity of , this factorises as

From the one-dimensional version of (3) (and the fact that is odd) we have

for some phase . The claim follows.

As one corollary of the functional equation, is a phase rotation of and thus is non-zero, so has degree exactly . The functional equation is then equivalent to the zeroes of being symmetric across the unit circle. In fact we have the stronger

Theorem 2 (Riemann hypothesis for Dirichlet -functions over function fields) Let be an odd Dirichlet character of modulus , and . Then all the zeroes of lie on the unit circle.

We derive this result from the Riemann hypothesis for curves over function fields below the fold.

In view of this theorem (and the fact that ), we may write

for some unitary matrix . It is possible to interpret as the action of the geometric Frobenius map on a certain cohomology group, but we will not do so here. The situation here is simpler than in the number field case because the factor arising from very small primes is now absent (in the function field setting there are no primes of size between and ).

We now let vary uniformly at random over all odd characters of modulus , and uniformly over , independently of ; we also make the distribution of the random variable conjugation invariant in . We use to denote the expectation with respect to this randomness. One can then ask what the limiting distribution of is in various regimes; we will focus in this post on the regime where is fixed and is being sent to infinity. In the spirit of the Sato-Tate conjecture, one should expect to converge in distribution to the circular unitary ensemble (CUE), that is to say Haar probability measure on . This may well be provable from Deligne’s “Weil II” machinery (in the spirit of this monograph of Katz and Sarnak), though I do not know how feasible this is or whether it has already been done in the literature; here we shall avoid using this machinery and study what partial results towards this CUE hypothesis one can make without it.

If one lets be the eigenvalues of (ordered arbitrarily), then we now have

and hence the are essentially elementary symmetric polynomials of the eigenvalues:

One can take log derivatives to conclude

On the other hand, as in the number field case one has the Dirichlet series expansion

where has sufficiently large real part, , and the von Mangoldt function is defined as when is the power of an irreducible and otherwise. We conclude the “explicit formula”

for , where

Similarly on inverting we have

Since we also have

for sufficiently large real part, where the Möbius function is equal to when is the product of distinct irreducibles, and otherwise, we conclude that the Möbius coefficients

are just the complete homogeneous symmetric polynomials of the eigenvalues:

One can then derive various algebraic relationships between the coefficients from various identities involving symmetric polynomials, but we will not do so here.

What do we know about the distribution of ? By construction, it is conjugation-invariant; from (2) it is also invariant with respect to the rotations for any phase . We also have the function field analogue of the Rudnick-Sarnak asymptotics:

Proposition 3 (Rudnick-Sarnak asymptotics) Let be nonnegative integers. If

then the moment

is equal to in the limit (holding fixed) unless for all , in which case it is equal to

Comparing this with Proposition 1 from this previous post, we thus see that all the low moments of are consistent with the CUE hypothesis (and also with the ACUE hypothesis, again by the previous post). The case of this proposition was essentially established by Andrade, Miller, Pratt, and Trinh.

Proof: We may assume the homogeneity relationship

since otherwise the claim follows from the invariance under phase rotation . By (6), the expression (9) is equal to

where

and consists of copies of for each , and similarly consists of copies of for each .

The polynomials and are monic of degree , which by hypothesis is less than the degree of , and thus they can only be scalar multiples of each other in if they are identical (in ). As such, we see that the average

vanishes unless , in which case this average is equal to . Thus the expression (9) simplifies to

There are at most choices for the product , and each one contributes to the above sum. All but of these choices are square-free, so by accepting an error of , we may restrict attention to square-free . This forces to all be irreducible (as opposed to powers of irreducibles); as is a unique factorisation domain, this forces and to be a permutation of . By the size restrictions, this then forces for all (if the above expression is to be anything other than ), and each is associated to possible choices of . Writing and then reinstating the non-squarefree possibilities for , we can thus write the above expression as

Using the prime number theorem , we obtain the claim.

Comparing this with Proposition 1 from this previous post, we thus see that all the low moments of are consistent with the CUE and ACUE hypotheses:

Corollary 4 (CUE statistics at low frequencies) Let be the eigenvalues of , permuted uniformly at random. Let be a linear combination of monomials where are integers with either or . Then

The analogue of the GUE hypothesis in this setting would be the CUE hypothesis, which asserts that the threshold here can be replaced by an arbitrarily large quantity. As far as I know this is not known even for (though, as mentioned previously, in principle one may be able to resolve such cases using Deligne’s proof of the Riemann hypothesis for function fields). Among other things, this would allow one to distinguish CUE from ACUE, since as discussed in the previous post, these two distributions agree when tested against monomials up to threshold , though not to .

Proof: By permutation symmetry we can take to be symmetric, and by linearity we may then take to be the symmetrisation of a single monomial . If then both expectations vanish due to the phase rotation symmetry, so we may assume that and . We can write this symmetric polynomial as a constant multiple of plus other monomials with a smaller value of . Since , the claim now follows by induction from Proposition 3 and Proposition 1 from the previous post.

Thus, for instance, for , the moment

is equal to

because all the monomials in are of the required form when . The latter expectation can be computed exactly (for any natural number ) using a formula

of Baker-Forrester and Keating-Snaith, thus for instance

and more generally

when , where are the integers

and more generally

(OEIS A039622). Thus we have

for if and is sufficiently slowly growing depending on . The CUE hypothesis would imply that that this formula also holds for higher . (The situation here is cleaner than in the number field case, in which the GUE hypothesis only suggests the correct lower bound for the moments rather than an asymptotic, due to the absence of the wildly fluctuating additional factor that is present in the Riemann zeta function model.)

Now we can recover the analogue of Montgomery’s work on the pair correlation conjecture. Consider the statistic

where

is some finite linear combination of monomials independent of . We can expand the above sum as

Assuming the CUE hypothesis, then by Example 3 of the previous post, we would conclude that

This is the analogue of Montgomery’s pair correlation conjecture. Proposition 3 implies that this claim is true whenever is supported on . If instead we assume the ACUE hypothesis (or the weaker Alternative Hypothesis that the phase gaps are non-zero multiples of ), one should instead have

for arbitrary ; this is the function field analogue of a recent result of Baluyot. In any event, since is non-negative, we unconditionally have the lower bound

if is non-negative for .

By applying (12) for various choices of test functions we can obtain various bounds on the behaviour of eigenvalues. For instance suppose we take the Fejér kernel

Then (12) applies unconditionally and we conclude that

The right-hand side evaluates to . On the other hand, is non-negative, and equal to when . Thus

The sum is at least , and is at least if is not a simple eigenvalue. Thus

and thus the expected number of simple eigenvalues is at least ; in particular, at least two thirds of the eigenvalues are simple asymptotically on average. If we had (12) without any restriction on the support of , the same arguments allow one to show that the expected proportion of simple eigenvalues is .

Suppose that the phase gaps in are all greater than almost surely. Let is non-negative and non-positive for outside of the arc . Then from (13) one has

so by taking contrapositives one can force the existence of a gap less than asymptotically if one can find with non-negative, non-positive for outside of the arc , and for which one has the inequality

By a suitable choice of (based on a minorant of Selberg) one can ensure this for for large; see Section 5 of these notes of Goldston. This is not the smallest value of currently obtainable in the literature for the number field case (which is currently , due to Goldston and Turnage-Butterbaugh, by a somewhat different method), but is still significantly less than the trivial value of . On the other hand, due to the compatibility of the ACUE distribution with Proposition 3, it is not possible to lower below purely through the use of Proposition 3.

In some cases it is possible to go beyond Proposition 3. Consider the mollified moment

where

for some coefficients . We can compute this moment in the CUE case:

Proposition 5 We have

Proof: From (5) one has

hence

where we suppress the dependence on the eigenvalues . Now observe the Pieri formula

where are the hook Schur polynomials

and we adopt the convention that vanishes for , or when and . Then also vanishes for . We conclude that

As the Schur polynomials are orthonormal on the unitary group, the claim follows.

The CUE hypothesis would then imply the corresponding mollified moment conjecture

(See this paper of Conrey, and this paper of Radziwill, for some discussion of the analogous conjecture for the zeta function, which is essentially due to Farmer.)

From Proposition 3 one sees that this conjecture holds in the range . It is likely that the function field analogue of the calculations of Conrey (based ultimately on deep exponential sum estimates of Deshouillers and Iwaniec) can extend this range to for any , if is sufficiently large depending on ; these bounds thus go beyond what is available from Proposition 3. On the other hand, as discussed in Remark 7 of the previous post, ACUE would also predict (14) for as large as , so the available mollified moment estimates are not strong enough to rule out ACUE. It would be interesting to see if there is some other estimate in the function field setting that can be used to exclude the ACUE hypothesis (possibly one that exploits the fact that GRH is available in the function field case?).

The alternative hypothesis for unitary matrices

In a recent post I discussed how the Riemann zeta function can be locally approximated by a polynomial, in the sense that for randomly chosen one has an approximation

where grows slowly with , and is a polynomial of degree . Assuming the Riemann hypothesis (as we will throughout this post), the zeroes of should all lie on the unit circle, and one should then be able to write as a scalar multiple of the characteristic polynomial of (the inverse of) a unitary matrix , which we normalise as

Here is some quantity depending on . We view as a random element of ; in the limit , the GUE hypothesis is equivalent to becoming equidistributed with respect to Haar measure on (also known as the Circular Unitary Ensemble, CUE; it is to the unit circle what the Gaussian Unitary Ensemble (GUE) is on the real line). One can also view as analogous to the “geometric Frobenius” operator in the function field setting, though unfortunately it is difficult at present to make this analogy any more precise (due, among other things, to the lack of a sufficiently satisfactory theory of the “field of one element“).

Taking logarithmic derivatives of (2), we have

and hence on taking logarithmic derivatives of (1) in the variable we (heuristically) have

Morally speaking, we have

so on comparing coefficients we expect to interpret the moments of as a finite Dirichlet series:

To understand the distribution of in the unitary group , it suffices to understand the distribution of the moments

where denotes averaging over , and . The GUE hypothesis asserts that in the limit , these moments converge to their CUE counterparts

where is now drawn uniformly in with respect to the CUE ensemble, and denotes expectation with respect to that measure.

The moment (6) vanishes unless one has the homogeneity condition

This follows from the fact that for any phase , has the same distribution as , where we use the number theory notation .

In the case when the degree is low, we can use representation theory to establish the following simple formula for the moment (6), as evaluated by Diaconis and Shahshahani:

Proposition 1 (Low moments in CUE model) If

then the moment (6) vanishes unless for all , in which case it is equal to

Another way of viewing this proposition is that for distributed according to CUE, the random variables are distributed like independent complex random variables of mean zero and variance , as long as one only considers moments obeying (8). This identity definitely breaks down for larger values of , so one only obtains central limit theorems in certain limiting regimes, notably when one only considers a fixed number of ‘s and lets go to infinity. (The paper of Diaconis and Shahshahani writes in place of , but I believe this to be a typo.)

Proof: Let be the left-hand side of (8). We may assume that (7) holds since we are done otherwise, hence

Our starting point is Schur-Weyl duality. Namely, we consider the -dimensional complex vector space

This space has an action of the product group : the symmetric group acts by permutation on the tensor factors, while the general linear group acts diagonally on the factors, and the two actions commute with each other. Schur-Weyl duality gives a decomposition

where ranges over Young tableaux of size with at most rows, is the -irreducible unitary representation corresponding to (which can be constructed for instance using Specht modules), and is the -irreducible polynomial representation corresponding with highest weight .

Let be a permutation consisting of cycles of length (this is uniquely determined up to conjugation), and let . The pair then acts on , with the action on basis elements given by

The trace of this action can then be computed as

where is the matrix coefficient of . Breaking up into cycles and summing, this is just

But we can also compute this trace using the Schur-Weyl decomposition (10), yielding the identity

where is the character on associated to , and is the character on associated to . As is well known, is just the Schur polynomial of weight applied to the (algebraic, generalised) eigenvalues of . We can specialise to unitary matrices to conclude that

and similarly

where consists of cycles of length for each . On the other hand, the characters are an orthonormal system on with the CUE measure. Thus we can write the expectation (6) as

Now recall that ranges over all the Young tableaux of size with at most rows. But by (8) we have , and so the condition of having rows is redundant. Hence now ranges over all Young tableaux of size , which as is well known enumerates all the irreducible representations of . One can then use the standard orthogonality properties of characters to show that the sum (12) vanishes if , are not conjugate, and is equal to divided by the size of the conjugacy class of (or equivalently, by the size of the centraliser of ) otherwise. But the latter expression is easily computed to be , giving the claim.

Example 2 We illustrate the identity (11) when , . The Schur polynomials are given as

where are the (generalised) eigenvalues of , and the formula (11) in this case becomes

The functions are orthonormal on , so the three functions are also, and their norms are , , and respectively, reflecting the size in of the centralisers of the permutations , , and respectively. If is instead set to say , then the terms now disappear (the Young tableau here has too many rows), and the three quantities here now have some non-trivial covariance.

Example 3 Consider the moment . For , the above proposition shows us that this moment is equal to . What happens for ? The formula (12) computes this moment as

where is a cycle of length in , and ranges over all Young tableaux with size and at most rows. The Murnaghan-Nakayama rule tells us that vanishes unless is a hook (all but one of the non-zero rows consisting of just a single box; this also can be interpreted as an exterior power representation on the space of vectors in whose coordinates sum to zero), in which case it is equal to (depending on the parity of the number of non-zero rows). As such we see that this moment is equal to . Thus in general we have

Now we discuss what is known for the analogous moments (5). Here we shall be rather non-rigorous, in particular ignoring an annoying “Archimedean” issue that the product of the ranges and is not quite the range but instead leaks into the adjacent range . This issue can be addressed by working in a “weak" sense in which parameters such as are averaged over fairly long scales, or by passing to a function field analogue of these questions, but we shall simply ignore the issue completely and work at a heuristic level only. For similar reasons we will ignore some technical issues arising from the sharp cutoff of to the range (it would be slightly better technically to use a smooth cutoff).

One can morally expand out (5) using (4) as

where , , and the integers are in the ranges

for and , and

for and . Morally, the expectation here is negligible unless

in which case the expecation is oscillates with magnitude one. In particular, if (7) fails (with some room to spare) then the moment (5) should be negligible, which is consistent with the analogous behaviour for the moments (6). Now suppose that (8) holds (with some room to spare). Then is significantly less than , so the multiplicative error in (15) becomes an additive error of . On the other hand, because of the fundamental integrality gap – that the integers are always separated from each other by a distance of at least – this forces the integers , to in fact be equal:

The von Mangoldt factors effectively restrict to be prime (the effect of prime powers is negligible). By the fundamental theorem of arithmetic, the constraint (16) then forces , and to be a permutation of , which then forces for all ._ For a given , the number of possible is then , and the expectation in (14) is equal to . Thus this expectation is morally

and using Mertens’ theorem this soon simplifies asymptotically to the same quantity in Proposition 1. Thus we see that (morally at least) the moments (5) associated to the zeta function asymptotically match the moments (6) coming from the CUE model in the low degree case (8), thus lending support to the GUE hypothesis. (These observations are basically due to Rudnick and Sarnak, with the degree case of pair correlations due to Montgomery, and the degree case due to Hejhal.)

With some rare exceptions (such as those estimates coming from “Kloostermania”), the moment estimates of Rudnick and Sarnak basically represent the state of the art for what is known for the moments (5). For instance, Montgomery’s pair correlation conjecture, in our language, is basically the analogue of (13) for , thus

for all . Montgomery showed this for (essentially) the range (as remarked above, this is a special case of the Rudnick-Sarnak result), but no further cases of this conjecture are known.

These estimates can be used to give some non-trivial information on the largest and smallest spacings between zeroes of the zeta function, which in our notation corresponds to spacing between eigenvalues of . One such method used today for this is due to Montgomery and Odlyzko and was greatly simplified by Conrey, Ghosh, and Gonek. The basic idea, translated to our random matrix notation, is as follows. Suppose is some random polynomial depending on of degree at most . Let denote the eigenvalues of , and let be a parameter. Observe from the pigeonhole principle that if the quantity

exceeds the quantity

then the arcs cannot all be disjoint, and hence there exists a pair of eigenvalues making an angle of less than ( times the mean angle separation). Similarly, if the quantity (18) falls below that of (19), then these arcs cannot cover the unit circle, and hence there exists a pair of eigenvalues making an angle of greater than times the mean angle separation. By judiciously choosing the coefficients of as functions of the moments , one can ensure that both quantities (18), (19) can be computed by the Rudnick-Sarnak estimates (or estimates of equivalent strength); indeed, from the residue theorem one can write (18) as

for sufficiently small , and this can be computed (in principle, at least) using (3) if the coefficients of are in an appropriate form. Using this sort of technology (translated back to the Riemann zeta function setting), one can show that gaps between consecutive zeroes of zeta are less than times the mean spacing and greater than times the mean spacing infinitely often for certain ; the current records are (due to Goldston and Turnage-Butterbaugh) and (due to Bui and Milinovich, who input some additional estimates beyond the Rudnick-Sarnak set, namely the twisted fourth moment estimates of Bettin, Bui, Li, and Radziwill, and using a technique based on Hall’s method rather than the Montgomery-Odlyzko method).

It would be of great interest if one could push the upper bound for the smallest gap below . The reason for this is that this would then exclude the Alternative Hypothesis that the spacing between zeroes are asymptotically always (or almost always) a non-zero half-integer multiple of the mean spacing, or in our language that the gaps between the phases of the eigenvalues of are nasymptotically always non-zero integer multiples of . The significance of this hypothesis is that it is implied by the existence of a Siegel zero (of conductor a small power of ); see this paper of Conrey and Iwaniec. (In our language, what is going on is that if there is a Siegel zero in which is very close to zero, then behaves like the Kronecker delta, and hence (by the Riemann-Siegel formula) the combined -function will have a polynomial approximation which in our language looks like a scalar multiple of , where and is a phase. The zeroes of this approximation lie on a coset of the roots of unity; the polynomial is a factor of this approximation and hence will also lie in this coset, implying in particular that all eigenvalue spacings are multiples of . Taking then gives the claim.)

Unfortunately, the known methods do not seem to break this barrier without some significant new input; already the original paper of Montgomery and Odlyzko observed this limitation for their particular technique (and in fact fall very slightly short, as observed in unpublished work of Goldston and of Milinovich). In this post I would like to record another way to see this, by providing an “alternative” probability distribution to the CUE distribution (which one might dub the Alternative Circular Unitary Ensemble (ACUE) which is indistinguishable in low moments in the sense that the expectation for this model also obeys Proposition 1, but for which the phase spacings are always a multiple of . This shows that if one is to rule out the Alternative Hypothesis (and thus in particular rule out Siegel zeroes), one needs to input some additional moment information beyond Proposition 1. It would be interesting to see if any of the other known moment estimates that go beyond this proposition are consistent with this alternative distribution. (UPDATE: it looks like they are, see Remark 7 below.)

To describe this alternative distribution, let us first recall the Weyl description of the CUE measure on the unitary group in terms of the distribution of the phases of the eigenvalues, randomly permuted in any order. This distribution is given by the probability measure

where

is the Vandermonde determinant; see for instance this previous blog post for the derivation of a very similar formula for the GUE distribution, which can be adapted to CUE without much difficulty. To see that this is a probability measure, first observe the Vandermonde determinant identity

where , denotes the dot product, and is the “long word”, which implies that (20) is a trigonometric series with constant term ; it is also clearly non-negative, so it is a probability measure. One can thus generate a random CUE matrix by first drawing using the probability measure (20), and then generating to be a random unitary matrix with eigenvalues .

For the alternative distribution, we first draw on the discrete torus (thus each is a root of unity) with probability density function

shift by a phase drawn uniformly at random, and then select to be a random unitary matrix with eigenvalues . Let us first verify that (21) is a probability density function. Clearly it is non-negative. It is the linear combination of exponentials of the form for . The diagonal contribution gives the constant function , which has total mass one. All of the other exponentials have a frequency that is not a multiple of , and hence will have mean zero on . The claim follows.

From construction it is clear that the matrix drawn from this alternative distribution will have all eigenvalue phase spacings be a non-zero multiple of . Now we verify that the alternative distribution also obeys Proposition 1. The alternative distribution remains invariant under rotation by phases, so the claim is again clear when (8) fails. Inspecting the proof of that proposition, we see that it suffices to show that the Schur polynomials with of size at most and of equal size remain orthonormal with respect to the alternative measure. That is to say,

when have size equal to each other and at most . In this case the phase in the definition of is irrelevant. In terms of eigenvalue measures, we are then reduced to showing that

By Fourier decomposition, it then suffices to show that the trigonometric polynomial does not contain any components of the form for some non-zero lattice vector . But we have already observed that is a linear combination of plane waves of the form for . Also, as is well known, is a linear combination of plane waves where is majorised by , and similarly is a linear combination of plane waves where is majorised by . So the product is a linear combination of plane waves of the form . But every coefficient of the vector lies between and , and so cannot be of the form for any non-zero lattice vector , giving the claim.

Example 4 If , then the distribution (21) assigns a probability of to any pair that is a permuted rotation of , and a probability of to any pair that is a permuted rotation of . Thus, a matrix drawn from the alternative distribution will be conjugate to a phase rotation of with probability , and to with probability .

A similar computation when gives conjugate to a phase rotation of with probability , to a phase rotation of or its adjoint with probability of each, and a phase rotation of with probability .

Remark 5 For large it does not seem that this specific alternative distribution is the only distribution consistent with Proposition 1 and which has all phase spacings a non-zero multiple of ; in particular, it may not be the only distribution consistent with a Siegel zero. Still, it is a very explicit distribution that might serve as a test case for the limitations of various arguments for controlling quantities such as the largest or smallest spacing between zeroes of zeta. The ACUE is in some sense the distribution that maximally resembles CUE (in the sense that it has the greatest number of Fourier coefficients agreeing) while still also being consistent with the Alternative Hypothesis, and so should be the most difficult enemy to eliminate if one wishes to disprove that hypothesis.

In some cases, even just a tiny improvement in known results would be able to exclude the alternative hypothesis. For instance, if the alternative hypothesis held, then is periodic in with period , so from Proposition 1 for the alternative distribution one has

which differs from (13) for any . (This fact was implicitly observed recently by Baluyot, in the original context of the zeta function.) Thus a verification of the pair correlation conjecture (17) for even a single with would rule out the alternative hypothesis. Unfortunately, such a verification appears to be on comparable difficulty with (an averaged version of) the Hardy-Littlewood conjecture, with power saving error term. (This is consistent with the fact that Siegel zeroes can cause distortions in the Hardy-Littlewood conjecture, as (implicitly) discussed in this previous blog post.)

Remark 6 One can view the CUE as normalised Lebesgue measure on (viewed as a smooth submanifold of ). One can similarly view ACUE as normalised Lebesgue measure on the (disconnected) smooth submanifold of consisting of those unitary matrices whose phase spacings are non-zero integer multiples of ; informally, ACUE is CUE restricted to this lower dimensional submanifold. As is well known, the phases of CUE eigenvalues form a determinantal point process with kernel (or one can equivalently take ; in a similar spirit, the phases of ACUE eigenvalues, once they are rotated to be roots of unity, become a discrete determinantal point process on those roots of unity with exactly the same kernel (except for a normalising factor of ). In particular, the -point correlation functions of ACUE (after this rotation) are precisely the restriction of the -point correlation functions of CUE after normalisation, that is to say they are proportional to .

Remark 7 One family of estimates that go beyond the Rudnick-Sarnak family of estimates are twisted moment estimates for the zeta function, such as ones that give asymptotics for

for some small even exponent (almost always or ) and some short Dirichlet polynomial ; see for instance this paper of Bettin, Bui, Li, and Radziwill for some examples of such estimates. The analogous unitary matrix average would be something like

where is now some random medium degree polynomial that depends on the unitary matrix associated to (and in applications will typically also contain some negative power of to cancel the corresponding powers of in ). Unfortunately such averages generally are unable to distinguish the CUE from the ACUE. For instance, if all the coefficients of involve products of traces of total order less than , then in terms of the eigenvalue phases , is a linear combination of plane waves where the frequencies have coefficients of magnitude less than . On the other hand, as each coefficient of is an elementary symmetric function of the eigenvalues, is a linear combination of plane waves where the frequencies have coefficients of magnitude at most . Thus is a linear combination of plane waves where the frequencies have coefficients of magnitude less than , and thus is orthogonal to the difference between the CUE and ACUE measures on the phase torus by the previous arguments. In other words, has the same expectation with respect to ACUE as it does with respect to CUE. Thus one can only start distinguishing CUE from ACUE if the mollifier has degree close to or exceeding , which corresponds to Dirichlet polynomials of length close to or exceeding , which is far beyond current technology for such moment estimates.

Remark 8 The GUE hypothesis for the zeta function asserts that the average

is equal to

for any and any test function , where is the Dyson sine kernel and are the ordinates of zeroes of the zeta function. This corresponds to the CUE distribution for . The ACUE distribution then corresponds to an “alternative gaussian unitary ensemble (AGUE)” hypothesis, in which the average (22) is instead predicted to equal a Riemann sum version of the integral (23):

This is a stronger version of the alternative hypothesis that the spacing between adjacent zeroes is almost always approximately a half-integer multiple of the mean spacing. I do not know of any known moment estimates for Dirichlet series that is able to eliminate this AGUE hypothesis (even assuming GRH). (UPDATE: These facts have also been independently observed in forthcoming work of Lagarias and Rodgers.)

PCMI lecture notes on random matrix theory

In July I will be spending a week at Park City, being one of the mini-course lecturers in the Graduate Summer School component of the Park City Summer Session on random matrices.  I have chosen to give some lectures on least singular values of random matrices, the circular law, and the Lindeberg exchange method in random matrix theory; this is a slightly different set of topics than I had initially advertised (which was instead about the Lindeberg exchange method and the local relaxation flow method), but after consulting with the other mini-course lecturers I felt that this would be a more complementary set of topics.  I have uploaded an draft of my lecture notes (some portion of which is derived from my monograph on the subject); as always, comments and corrections are welcome.

[Update, June 23: notes revised and reformatted to PCMI format. -T.]

 

[Update, Mar 19 2018: further revision. -T.]

Random matrices have simple spectrum

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):

Theorem 1 With probability , has simple eigenvalues.

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.)

Some notes on Bakry-Emery theory

[These are notes intended mostly for myself, as these topics are useful in random matrix theory, but may be of interest to some readers also. -T.]

One of the most fundamental partial differential equations in mathematics is the heat equation

where is a scalar function of both time and space, and is the Laplacian . For the purposes of this post, we will ignore all technical issues of regularity and decay, and always assume that the solutions to equations such as (1) have all the regularity and decay in order to justify all formal operations such as the chain rule, integration by parts, or differentiation under the integral sign. The factor of in the definition of the heat propagator is of course an arbitrary normalisation, chosen for some minor technical reasons; one can certainly continue the discussion below with other choices of normalisations if desired.

In probability theory, this equation takes on particular significance when is restricted to be non-negative, and furthermore to be a probability measure at each time, in the sense that

for all . (Actually, it suffices to verify this constraint at time , as the heat equation (1) will then preserve this constraint.) Indeed, in this case, one can interpret as the probability distribution of a Brownian motion

where is a stochastic process with initial probability distribution ; see for instance this previous blog post for more discussion.

A model example of a solution to the heat equation to keep in mind is that of the fundamental solution

defined for any , which represents the distribution of Brownian motion of a particle starting at the origin at time . At time , represents an -valued random variable, each coefficient of which is an independent random variable of mean zero and variance . (As , converges in the sense of distributions to a Dirac mass at the origin.)

The heat equation can also be viewed as the gradient flow for the Dirichlet form

since one has the integration by parts identity

for all smooth, rapidly decreasing , which formally implies that is (half of) the negative gradient of the Dirichlet energy with respect to the inner product. Among other things, this implies that the Dirichlet energy decreases in time:

For instance, for the fundamental solution (3), one can verify for any time that

(assuming I have not made a mistake in the calculation). In a similar spirit we have

Since is non-negative, the formula (6) implies that is integrable in time, and in particular we see that converges to zero as , in some averaged sense at least; similarly, (8) suggests that also converges to zero. This suggests that converges to a constant function; but as is also supposed to decay to zero at spatial infinity, we thus expect solutions to the heat equation in to decay to zero in some sense as . However, the decay is only expected to be polynomial in nature rather than exponential; for instance, the solution (3) decays in the norm like .

Since , we also observe the basic cancellation property

for any function .

There are other quantities relating to that also decrease in time under heat flow, particularly in the important case when is a probability measure. In this case, it is natural to introduce the entropy

Thus, for instance, if is the uniform distribution on some measurable subset of of finite measure , the entropy would be . Intuitively, as the entropy decreases, the probability distribution gets wider and flatter. For instance, in the case of the fundamental solution (3), one has for any , reflecting the fact that is approximately uniformly distributed on a ball of radius (and thus of measure ).

A short formal computation shows (if one assumes for simplicity that is strictly positive, which is not an unreasonable hypothesis, particularly in view of the strong maximum principle) using (9), (5) that

where is the square root of . For instance, if is the fundamental solution (3), one can check that (note that this is a significantly cleaner formula than (7)!).

In particular, the entropy is decreasing, which corresponds well to one’s intuition that the heat equation (or Brownian motion) should serve to spread out a probability distribution over time.

Actually, one can say more: the rate of decrease of the entropy is itself decreasing, or in other words the entropy is convex. I do not have a satisfactorily intuitive reason for this phenomenon, but it can be proved by straightforward application of basic several variable calculus tools (such as the chain rule, product rule, quotient rule, and integration by parts), and completing the square. Namely, by using the chain rule we have

valid for for any smooth function , we see from (1) that

and thus (again assuming that , and hence , is strictly positive to avoid technicalities)

We thus have

It is now convenient to compute using the Einstein summation convention to hide the summation over indices . We have

and

By integration by parts and interchanging partial derivatives, we may write the first integral as

and from the quotient and product rules, we may write the second integral as

Gathering terms, completing the square, and making the summations explicit again, we see that

and so in particular is always decreasing.

The above identity can also be written as

Exercise 1 Give an alternate proof of the above identity by writing , and deriving the equation for .

It was observed in a well known paper of Bakry and Emery that the above monotonicity properties hold for a much larger class of heat flow-type equations, and lead to a number of important relations between energy and entropy, such as the log-Sobolev inequality of Gross and of Federbush, and the hypercontractivity inequality of Nelson; we will discuss one such family of generalisations (or more precisely, variants) below the fold.

Random matrices: Universality of local spectral statistics of non-Hermitian matrices

Van Vu and I have just uploaded to the arXiv our paper “Random matrices: Universality of local spectral statistics of non-Hermitian matrices“. The main result of this paper is a “Four Moment Theorem” that establishes universality for local spectral statistics of non-Hermitian matrices with independent entries, under the additional hypotheses that the entries of the matrix decay exponentially, and match moments with either the real or complex gaussian ensemble to fourth order. This is the non-Hermitian analogue of a long string of recent results establishing universality of local statistics in the Hermitian case (as discussed for instance in this recent survey of Van and myself, and also in several other places).

The complex case is somewhat easier to describe. Given a (non-Hermitian) random matrix ensemble of matrices, one can arbitrarily enumerate the (geometric) eigenvalues as , and one can then define the -point correlation functions to be the symmetric functions such that

In the case when is drawn from the complex gaussian ensemble, so that all the entries are independent complex gaussians of mean zero and variance one, it is a classical result of Ginibre that the asymptotics of near some point as and is fixed are given by the determinantal rule

for and

for , where is the reproducing kernel

(There is also an asymptotic for the boundary case , but it is more complicated to state.) In particular, we see that for almost every , which is a manifestation of the well-known circular law for these matrices; but the circular law only captures the macroscopic structure of the spectrum, whereas the asymptotic (1) describes the microscopic structure.

Our first main result is that the asymptotic (1) for also holds (in the sense of vague convergence) when is a matrix whose entries are independent with mean zero, variance one, exponentially decaying tails, and which all match moments with the complex gaussian to fourth order. (Actually we prove a stronger result than this which is valid for all bounded and has more uniform bounds, but is a bit more technical to state.) An analogous result is also established for real gaussians (but now one has to separate the correlation function into components depending on how many eigenvalues are real and how many are strictly complex; also, the limiting distribution is more complicated, being described by Pfaffians rather than determinants). Among other things, this allows us to partially extend some known results on complex or real gaussian ensembles to more general ensembles. For instance, there is a central limit theorem of Rider which establishes a central limit theorem for the number of eigenvalues of a complex gaussian matrix in a mesoscopic disk; from our results, we can extend this central limit theorem to matrices that match the complex gaussian ensemble to fourth order, provided that the disk is small enough (for technical reasons, our error bounds are not strong enough to handle large disks). Similarly, extending some results of Edelman-Kostlan-Shub and of Forrester-Nagao, we can show that for a matrix matching the real gaussian ensemble to fourth order, the number of real eigenvalues is with probability for some absolute constant .

There are several steps involved in the proof. The first step is to apply the Girko Hermitisation trick to replace the problem of understanding the spectrum of a non-Hermitian matrix, with that of understanding the spectrum of various Hermitian matrices. The two identities that realise this trick are, firstly, Jensen’s formula

that relates the local distribution of eigenvalues to the log-determinants , and secondly the elementary identity

that relates the log-determinants of to the log-determinants of the Hermitian matrices

The main difficulty is then to obtain concentration and universality results for the Hermitian log-determinants . This turns out to be a task that is analogous to the task of obtaining concentration for Wigner matrices (as we did in this recent paper), as well as central limit theorems for log-determinants of Wigner matrices (as we did in this other recent paper). In both of these papers, the main idea was to use the Four Moment Theorem for Wigner matrices (which can now be proven relatively easily by a combination of the local semi-circular law and resolvent swapping methods), combined with (in the latter paper) a central limit theorem for the gaussian unitary ensemble (GUE). This latter task was achieved by using the convenient Trotter normal form to tridiagonalise a GUE matrix, which has the effect of revealing the determinant of that matrix as the solution to a certain linear stochastic difference equation, and one can analyse the distribution of that solution via such tools as the martingale central limit theorem.

The matrices are somewhat more complicated than Wigner matrices (for instance, the semi-circular law must be replaced by a distorted Marchenko-Pastur law), but the same general strategy works to obtain concentration and universality for their log-determinants. The main new difficulty that arises is that the analogue of the Trotter norm for gaussian random matrices is not tridiagonal, but rather Hessenberg (i.e. upper-triangular except for the lower diagonal). This ultimately has the effect of expressing the relevant determinant as the solution to a nonlinear stochastic difference equation, which is a bit trickier to solve for. Fortunately, it turns out that one only needs good lower bounds on the solution, as one can use the second moment method to upper bound the determinant and hence the log-determinant (following a classical computation of Turan). This simplifies the analysis on the equation somewhat.

While this result is the first local universality result in the category of random matrices with independent entries, there are still two limitations to the result which one would like to remove. The first is the moment matching hypotheses on the matrix. Very recently, one of the ingredients of our paper, namely the local circular law, was proved without moment matching hypotheses by Bourgade, Yau, and Yin (provided one stays away from the edge of the spectrum); however, as of this time of writing the other main ingredient – the universality of the log-determinant – still requires moment matching. (The standard tool for obtaining universality without moment matching hypotheses is the heat flow method (and more specifically, the local relaxation flow method), but the analogue of Dyson Brownian motion in the non-Hermitian setting appears to be somewhat intractible, being a coupled flow on both the eigenvalues and eigenvectors rather than just on the eigenvalues alone.)

Random matrices: Sharp concentration of eigenvalues

Van Vu and I have just uploaded to the arXiv our paper Random matrices: Sharp concentration of eigenvalues, submitted to the Electronic Journal of Probability. As with many of our previous papers, this paper is concerned with the distribution of the eigenvalues of a random Wigner matrix (such as a matrix drawn from the Gaussian Unitary Ensemble (GUE) or Gaussian Orthogonal Ensemble (GOE)). To simplify the discussion we shall mostly restrict attention to the bulk of the spectrum, i.e. to eigenvalues with for some fixed , although analogues of most of the results below have also been obtained at the edge of the spectrum.

If we normalise the entries of the matrix to have mean zero and variance , then in the asymptotic limit , we have the Wigner semicircle law, which asserts that the eigenvalues are asymptotically distributed according to the semicircular distribution , where

An essentially equivalent way of saying this is that for large , we expect the eigenvalue of to stay close to the classical location , defined by the formula

In particular, from the Wigner semicircle law it can be shown that asymptotically almost surely, one has

for all .

In the modern study of the spectrum of Wigner matrices (and in particular as a key tool in establishing universality results), it has become of interest to improve the error term in (1) as much as possible. A typical early result in this direction was by Bai, who used the Stieltjes transform method to obtain polynomial convergence rates of the shape for some absolute constant ; see also the subsequent papers of Alon-Krivelevich-Vu and of of Meckes, who were able to obtain such convergence rates (with exponentially high probability) by using concentration of measure tools, such as Talagrand’s inequality. On the other hand, in the case of the GUE ensemble it is known (by this paper of Gustavsson) that has variance comparable to in the bulk, so that the optimal error term in (1) should be about . (One may think that if one wanted bounds on (1) that were uniform in , one would need to enlarge the error term further, but this does not appear to be the case, due to strong correlations between the ; note for instance this recent result of Ben Arous and Bourgarde that the largest gap between eigenvalues in the bulk is typically of order .)

A significant advance in this direction was achieved by Erdos, Schlein, and Yau in a series of papers where they used a combination of Stieltjes transform and concentration of measure methods to obtain local semicircle laws which showed, among other things, that one had asymptotics of the form

with exponentially high probability for intervals in the bulk that were as short as for some , where is the number of eigenvalues. These asymptotics are consistent with a good error term in (1), and are already sufficient for many applications, but do not quite imply a strong concentration result for individual eigenvalues (basically because they do not preclude long-range or “secular” shifts in the spectrum that involve large blocks of eigenvalues at mesoscopic scales). Nevertheless, this was rectified in a subsequent paper of Erdos, Yau, and Yin, which roughly speaking obtained a bound of the form

in the bulk with exponentially high probability, for Wigner matrices obeying some exponential decay conditions on the entries. This was achieved by a rather delicate high moment calculation, in which the contribution of the diagonal entries of the resolvent (whose average forms the Stieltjes transform) was shown to mostly cancel each other out.

As the GUE computations show, this concentration result is sharp up to the quasilogarithmic factor . The main result of this paper is to improve the concentration result to one more in line with the GUE case, namely

with exponentially high probability (see the paper for a more precise statement of results). The one catch is that an additional hypothesis is required, namely that the entries of the Wigner matrix have vanishing third moment. We also obtain similar results for the edge of the spectrum (but with a different scaling).

Our arguments are rather different from those of Erdos, Yau, and Yin, and thus provide an alternate approach to establishing eigenvalue concentration. The main tool is the Lindeberg exchange strategy, which is also used to prove the Four Moment Theorem (although we do not directly invoke the Four Moment Theorem in our analysis). The main novelty is that this exchange strategy is now used to establish large deviation estimates (i.e. exponentially small tail probabilities) rather than universality of the limiting distribution. Roughly speaking, the basic point is as follows. The Lindeberg exchange strategy seeks to compare a function of many independent random variables with the same function of a different set of random variables (which match moments with the original set of variables to some order, such as to second or fourth order) by exchanging the random variables one at a time. Typically, one tries to upper bound expressions such as

for various smooth test functions , by performing a Taylor expansion in the variable being swapped and taking advantage of the matching moment hypotheses. In previous implementations of this strategy, was a bounded test function, which allowed one to get control of the bulk of the distribution of , and in particular in controlling probabilities such as

for various thresholds and , but did not give good control on the tail as the error terms tended to be polynomially decaying in rather than exponentially decaying. However, it turns out that one can modify the exchange strategy to deal with moments such as

for various moderately large (e.g. of size comparable to ), obtaining results such as

after performing all the relevant exchanges. As such, one can then use large deviation estimates on to deduce large deviation estimates on .

In this paper we also take advantage of a simplification, first noted by Erdos, Yau, and Yin, that Four Moment Theorems become somewhat easier to prove if one works with resolvents (and the closely related Stieltjes transform ) rather than with individual eigenvalues, as the Taylor expansion of resolvents are very simple (essentially being a Neumann series). The relationship between the Stieltjes transform and the location of individual eigenvalues can be seen by taking advantage of the identity

for any energy level , which can be verified from elementary calculus. (In practice, we would truncate near zero and near infinity to avoid some divergences, but this is a minor technicality.) As such, a concentration result for the Stieltjes transform can be used to establish an analogous concentration result for the eigenvalue counting functions , which in turn can be used to deduce concentration results for individual eigenvalues by some basic combinatorial manipulations.

Random matrices: The Four Moment Theorem for Wigner ensembles

Van Vu and I have just uploaded to the arXiv our short survey article, “Random matrices: The Four Moment Theorem for Wigner ensembles“, submitted to the MSRI book series, as part of the proceedings on the MSRI semester program on random matrix theory from last year.  This is a highly condensed version (at 17 pages) of a much longer survey (currently at about 48 pages, though not completely finished) that we are currently working on, devoted to the recent advances in understanding the universality phenomenon for spectral statistics of Wigner matrices.  In this abridged version of the survey, we focus on a key tool in the subject, namely the Four Moment Theorem which roughly speaking asserts that the statistics of a Wigner matrix depend only on the first four moments of the entries.  We give a sketch of proof of this theorem, and two sample applications: a central limit theorem for individual eigenvalues of a Wigner matrix (extending a result of Gustavsson in the case of GUE), and the verification of a conjecture of Wigner, Dyson, and Mehta on the universality of the asymptotic k-point correlation functions even for discrete ensembles (provided that we interpret convergence in the vague topology sense).

For reasons of space, this paper is very far from an exhaustive survey even of the narrow topic of universality for Wigner matrices, but should hopefully be an accessible entry point into the subject nevertheless.

The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices

Van Vu and I have just uploaded to the arXiv our paper “The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices“, submitted to the Proceedings of the National Academy of Sciences. This short note concerns the convergence of the -point correlation functions of Wigner matrices in the bulk to the Dyson -point functions, a statement conjectured by Wigner, Dyson, and Mehta. Thanks to the results of Erdös, Peche, Ramirez, Schlein, Vu, Yau, and myself, this conjecture has now been established for all Wigner matrices (assuming a finite moment condition on the entries), but only if one uses a quite weak notion of convergence, namely averaged vague convergence in which one averages in the energy parameter . The main purpose of this note is to observe that by combining together existing results in the literature, one can improve the convergence to vague convergence (which is the natural notion of convergence in the discrete setting); and furthermore, if one assumes some regularity and decay conditions on the coefficient distribution, one can improve the convergence further to local convergence.

More precisely, let be an Wigner matrix – a random Hermitian matrix whose off-diagonal elements for are iid with mean zero and variance (and whose diagonal elements also obey similar hypotheses, which we omit here). For simplicity, we also assume that the real and imaginary parts of are also iid (as is the case for instance for the Gaussian Unitary Ensemble (GUE)). The eigenvalues of such a matrix are known to be asymptotically distributed accordingly to the Wigner semicircular distribution , where

In particular, this suggests that at any energy level in the bulk of the spectrum, the average eigenvalue spacing should be about . It is then natural to introduce the normalised -point correlation function

for any distinct reals and , where is the event that there is an eigenvalue in each of the intervals for each . (This definition is valid when the Wigner ensemble is continuous; for discrete ensembles, one can define instead in a distributional sense.)

The Wigner-Dyson-Mehta conjecture asserts that converges (in various senses) as to the Dyson -point function

where is the Dyson sine kernel. This conjecture was verified first for the GUE (with a quite strong notion of convergence, namely local uniform convergence) by Dyson, using an explicit formula for in the GUE case due to Gaudin and Mehta. Later results of Johansson, Erdos-Ramirez-Schlein-Yau, Erdos-Peche-Ramirez-Schlein-Yau, and Vu and myself, extended these results to increasingly wider ranges of Wigner matrices, but in the context of either weak convergence (which means that

for any , compactly supported function ), or the slightly weaker notion of vague convergence (which is the same as weak convergence, except that the function is also required to be continuous).

In a joint paper of Erdos, Ramirez, Schlein, Vu, Yau, and myself, we established the Wigner-Dyson-Mehta conjecture for all Wigner matrices (assuming only an exponential decay condition on the entries), but using a quite weak notion of convergence, namely averaged vague convergence, which allows for averaging in the energy parameter. Specifically, we showed that

Subsequently, Erdos, Schlein, and Yau introduced the powerful local relaxation flow method, which achieved a simpler proof of the same result which also generalised to other ensembles beyond the Wigner case. However, for technical reasons, this method was restricted to establishing averaged vague convergence only.

In the current paper, we show that by combining the argument of Erdos, Ramirez, Schlein, Vu, Yau, and myself with some more recent technical results, namely the relaxation of the exponential decay condition in the four moment theorem to a finite moment condition (established by Vu and myself) and a strong eigenvalue localisation bound of Erdos, Yau, and Yin, one can upgrade the averaged vague convergence to vague convergence, and handle all Wigner matrices that assume a finite moment condition. Vague convergence is the most natural notion of convergence for discrete random matrix ensembles; for such ensembles, the correlation function is a discrete measure, and so one does not expect convergence to a continuous limit in any stronger sense than the vague sense. Also, by carefully inspecting the earlier argument of Erdos, Peche, Ramirez, Schlein, and Yau, we were able to establish convergence in the stronger local sense once one assumed some regularity and positivity condition on the underlying coefficient distribution. These are somewhat modest and technical improvements over previous work on the Wigner-Dyson-Mehta conjecture, but they help to clarify and organise the profusion of results in this area, which are now reaching a fairly definitive form.

It may well be possible to go beyond local convergence in the case of smooth ensembles, for instance establishing local uniform convergence; this was recently accomplished in the case by Maltsev and Schlein. Indeed one may optimistically expect to even have convergence in the local smooth topology, which would basically be the strongest convergence one could hope for.

Outliers in the spectrum of iid matrices with bounded rank perturbations

I’ve just uploaded to the arXiv my paper “Outliers in the spectrum of iid matrices with bounded rank perturbations“, submitted to Probability Theory and Related Fields. This paper is concerned with outliers to the circular law for iid random matrices. Recall that if is an matrix whose entries are iid complex random variables with mean zero and variance one, then the complex eigenvalues of the normalised matrix will almost surely be distributed according to the circular law distribution in the limit . (See these lecture notes for further discussion of this law.)

The circular law is also stable under bounded rank perturbations: if is a deterministic rank matrix of polynomial size (i.e. of operator norm ), then the circular law also holds for (this is proven in a paper of myself, Van Vu, and Manjunath Krisnhapur). In particular, the bulk of the eigenvalues (i.e. of the eigenvalues) will lie inside the unit disk .

However, this leaves open the possibility for one or more outlier eigenvalues that lie significantly outside the unit disk; the arguments in the paper cited above give some upper bound on the number of such eigenvalues (of the form for some absolute constant ) but does not exclude them entirely. And indeed, numerical data shows that such outliers can exist for certain bounded rank perturbations.

In this paper, some results are given as to when outliers exist, and how they are distributed. The easiest case is of course when there is no bounded rank perturbation: . In that case, an old result of Bai and Yin and of Geman shows that the spectral radius of is almost surely , thus all eigenvalues will be contained in a neighbourhood of the unit disk, and so there are no significant outliers. The proof is based on the moment method.

Now we consider a bounded rank perturbation which is nonzero, but which has a bounded operator norm: . In this case, it turns out that the matrix will have outliers if the deterministic component has outliers. More specifically (and under the technical hypothesis that the entries of have bounded fourth moment), if is an eigenvalue of with , then (for large enough), will almost surely have an eigenvalue at , and furthermore these will be the only outlier eigenvalues of .

Thus, for instance, adding a bounded nilpotent low rank matrix to will not create any outliers, because the nilpotent matrix only has eigenvalues at zero. On the other hand, adding a bounded Hermitian low rank matrix will create outliers as soon as this matrix has an operator norm greater than .

When I first thought about this problem (which was communicated to me by Larry Abbott), I believed that it was quite difficult, because I knew that the eigenvalues of non-Hermitian matrices were quite unstable with respect to general perturbations (as discussed in this previous blog post), and that there were no interlacing inequalities in this case to control bounded rank perturbations (as discussed in this post). However, as it turns out I had arrived at the wrong conclusion, especially in the exterior of the unit disk in which the resolvent is actually well controlled and so there is no pseudospectrum present to cause instability. This was pointed out to me by Alice Guionnet at an AIM workshop last week, after I had posed the above question during an open problems session. Furthermore, at the same workshop, Percy Deift emphasised the point that the basic determinantal identity

for matrices and matrices was a particularly useful identity in random matrix theory, as it converted problems about large () matrices into problems about small () matrices, which was particularly convenient in the regime when and was fixed. (Percy was speaking in the context of invariant ensembles, but the point is in fact more general than this.)

From this, it turned out to be a relatively simple manner to transform what appeared to be an intractable matrix problem into quite a well-behaved matrix problem for bounded . Specifically, suppose that had rank , so that one can factor for some (deterministic) matrix and matrix . To find an eigenvalue of , one has to solve the characteristic polynomial equation

This is an determinantal equation, which looks difficult to control analytically. But we can manipulate it using (1). If we make the assumption that is outside the spectrum of (which we can do as long as is well away from the unit disk, as the unperturbed matrix has no outliers), we can divide by to arrive at

Now we apply the crucial identity (1) to rearrange this as

The crucial point is that this is now an equation involving only a determinant, rather than an one, and is thus much easier to solve. The situation is particularly simple for rank one perturbations

in which case the eigenvalue equation is now just a scalar equation

that involves what is basically a single coefficient of the resolvent . (It is also an instructive exercise to derive this eigenvalue equation directly, rather than through (1).) There is by now a very well-developed theory for how to control such coefficients (particularly for in the exterior of the unit disk, in which case such basic tools as Neumann series work just fine); in particular, one has precise enough control on these coefficients to obtain the result on outliers mentioned above.

The same method can handle some other bounded rank perturbations. One basic example comes from looking at iid matrices with a non-zero mean and variance ; this can be modeled by where is the unit vector . Here, the bounded rank perturbation has a large operator norm (equal to ), so the previous result does not directly apply. Nevertheless, the self-adjoint nature of the perturbation has a stabilising effect, and I was able to show that there is still only one outlier, and that it is at the expected location of .

If one moves away from the case of self-adjoint perturbations, though, the situation changes. Let us now consider a matrix of the form , where is a randomised version of , e.g. , where the are iid Bernoulli signs; such models were proposed recently by Rajan and Abbott as a model for neural networks in which some nodes are excitatory (and give columns with positive mean) and some are inhibitory (leading to columns with negative mean). Despite the superficial similarity with the previous example, the outlier behaviour is now quite different. Instead of having one extremely large outlier (of size ) at an essentially deterministic location, we now have a number of eigenvalues of size , scattered according to a random process. Indeed, (in the case when the entries of were real and bounded) I was able to show that the outlier point process converged (in the sense of converging -point correlation functions) to the zeroes of a random Laurent series

where are iid real Gaussians. This is basically because the coefficients of the resolvent have a Neumann series whose coefficients enjoy a central limit theorem.

On the other hand, as already observed numerically (and rigorously, in the gaussian case) by Rajan and Abbott, if one projects such matrices to have row sum zero, then the outliers all disappear. This can be explained by another appeal to (1); this projection amounts to right-multiplying by the projection matrix to the zero-sum vectors. But by (1), the non-zero eigenvalues of the resulting matrix are the same as those for . Since annihilates , we thus see that in this case the bounded rank perturbation plays no role, and the question reduces to obtaining a circular law with no outliers for . As it turns out, this can be done by invoking the machinery of Van Vu and myself that we used to prove the circular law for various random matrix models.