random matrices and the Riemann zeta function
There exist important conjectures which relate the statistical
behaviour of the nontrivial zeros of the Riemann
zeta function to the statistical behaviour of the eigenvalues of
large random matrices.
Although an individual matrix with fixed entries cannot meaningfully
be said to be 'random',
it is possible to precisely define 'random matrix ensembles' in terms
of probability distributions.
Most relevant here is the Gaussian unitary ensemble, or GUE.
Two equivalent ways of defining this for a given matrix
dimension N are given here:
(1) M.L. Mehta (in his seminal
book Random
Matrices) starts with the space of all N x N Hermitian
matrices, and introduces a probability density P on the space.
Two conditions are given which P must satisfy, including
invariance under unitary conjugation transformations
H > U^{1}HU on the space (here U
is a fixed N x N unitary matrix).
It turns out that the conditions given force P(H) to take
the form
Note the 'Gaussian' character of this
distribution.
Accepting this definition, we see that there really is no such condition as
being 'a GUE matrix'. The GUE consists of all Hermitian N x N
matrices, but they are 'weighted' according to the probability density
P.
I sometimes find it helpful to think of the GUE as 'a random
matrix generator' which produces N x N Hermitian matrices
with a probabilistic bias based on the function P.
Any Hermitian matrix of dimension N can be produced, but
some will occur more frequently than others. We are going
to be looking at statistical tendencies in the behaviour of the eigenvalues
of the 'output' of the generator. We will be particularly interested
in what happens to these tendencies when we let N tend to infinity.
(2) An equivalent definition is provided by Andrew
Odlyzko in
his unpublished 1992 paper "The 10^{20}th
zero of the Riemann zeta function and 175 million of its neighbors":
Despite the comments above, the idea is to construct a sort of 'archetypal GUE matrix'
built from matrix entries involving Gaussian random variables:
We note that the diagonal elements of a Hermitian matrix must be
real, and that the (complex) abovediagonal elements determine the (complex) belowdiagonal
elements via complex conjugation. Hence we can parametrise our
N x N Hermitian matrix in terms of a fixed number of
real variables (one for each diagonal element, and two for each abovediagonal
element  a total of N^{2} parameters).
Having introduced a 2^{1/2} scaling factor
into the real variables representing diagonal elements (for technical reasons), all of the
real parameters are assumed
to be independent standard normal variables. Recall that this means they
distribute according to the Gaussian distribution with zero mean
and variance 1.
Again, it's important to stress the fact that any Hermitian
N x N matrix could be said to 'belong' to the GUE according to this
(equivalent) definition, but the involvement
of probability distributions guarantees that some are 'more likely
to show up' than others, according to a very particular law of
distribution.
A couple of excellently composed and wellreferenced introductions to the
relationship between the GUE and the Riemann zeta function:
Steven Finch's notes "GUE
hypothesis regarding zeta function spacings"
D. Rockmore and D.L. Snell, "Chance in the Primes, Part III",
Chance
News
These are complemented by Dan Bump's
commentary on the Gaussian Unitary Ensemble Hypothesis, which
is a more casual, less technical affair explaining the connections
with physics and the Riemann Hypothesis.
I would recommend these three sources (and Mehta's book) for a thorough
account of these matters. The commentary immediately below stresses a few points
which may help to assist in understanding the main ideas.
Asymptotically, the locations of the nontrivial zeros of the
Riemann zeta function distribute
in the critical strip according to a fairly simple logarithmic
law, much like the primes do along the real line. Whereas the primes tend to become
increasingly spaced out at a logarithmic rate, the zeta zeros tend
to become more dense.
In order to undertake the most useful statistical analysis of
the zeta zeros, it is necessary to 'normalise' them, by applying
an appropriate rescaling. The idea is to cause the average
spacing between consecutive normalised zeros to equal 1. We can then consider the distribution
of these normalised spacings (a countably infinite set of positive values
which tend to cluster around the value 1).
Similarly, given a Hermitian matrix of (large) dimension N,
as we ascend the spectrum of eigenvalues we find an tendency towards
increasing density between larger eigenvalues. This effect
can be cancelled by rescaling to produce 'normalised eigenvalues' whose
'average consecutive spacing'
is equal to 1. In this way, we can compute the distribution of
spacings between consecutive normalised eigenvalues over the entire GUE of N
x N Hermitian matrices. Remember that although this computation
will involve all N x N Hermitian matrices,
there is a 'weighting' so some will have a more significant effect
than others.
In his aforementioned paper, Odlyzko sums up the situation:
"The basic results about distribution of GUE eigenvalues are
completely rigorous. However, they do have many gaps. One of them is
that the results are obtained by averaging over the full ensemble of
GUE matrices. It is conjectured that if one considers a large random GUE
matrix, the distribution of its eigenvalues will be close to that of the
entire ensemble with high probability. Although numerical calculations
confirm this conjecture, there is no proof of it."
The GUE Hypothesis, also known as the OdlyzkoMontgomery
Law, states that these two 'consecutive normalised spacing' distributions
(zeta zeros and GUE eigenvalues) are identical. In other
words, the nontrivial zeros of the Riemann zeta function and the
eigenvalues of GUE matrices share a particular 'statistical fingerprint'.
This hypothesis was inspired by some theoretical work of Montgomery, and backed up
by a huge body of experimental evidence provided by Odlyzko.
[Note: In Steve
Finch's notes it is pointed out that there is a slight technicality involving
the use of only the eigenvalues in the central part or "bulk" of GUE spectra.]
Previous evidence of the nontrivial zeros of the Riemann zeta function
behaving like eigenvalues appeared in
the early 1950's when Selberg produced his trace formula.
This possiblity, that the zeta zeros are 'spectral in nature',
is particularly attractive to those trying to prove
the Riemann Hypothesis via the suggestion of Hilbert and Pólya.
It is possible to give an explicit expression for this probability
distribution which both GUE eigenvalue spacings and zeta zero spacings
are conjectured to follow:
There is a sequence of distributions {p(k,u)}
of which this is the first. These can all be expressed as power series,
and are such that
Here w^{(k)} denotes
the k^{th} normalised eigenvalue (or zeta zero) to be greater than a given
normalised eigenvalue (or zeta zero) w. The probability that w^{(k)}
lies at a distance beyond w between a
given pair of limits is equal to the area under the graph of p(k1,u)
between the same limits.
Note that w^{(1)} is the first normalised eigenvalue (or zeta zero) to be
greater than w, hence w^{(1)}  w
is the distance between consecutive normalised eigenvalues (or zeta zeros)
which is what we have been discussing above. It therefore makes sense
that
A further prediction, which Odlyzko refers to as the Montgomery pair
correlation conjecture, deals with another statistical similarity
seemingly shared by the zeta zeros and the GUE eigenvalues.
Here we see a graph of the function
The idea, essentially, is this. Given a normalised zeta zero, and a pair
of positive values a, b with a < b, we
can consider the number of zeta zeros whose 'heights' above the given zero
are between a and b.
Montgomery has conjectured that the mean value of such quantity (taken over
the whole set of nontrivial zeta zeros) will equal the area under the
graph of g(r) between limits a and b (i.e. the
integral of g between these limits).
Exactly analogous behaviour had been discovered in the context of random matrix theory by
Freeman Dyson, who had been investigating
the possibility of using random matrices to model the behaviour of
heavy nuclei. Montgomery and Dyson
coincidentally became aware of the similarity in each others' work in 1972 while talking
informally over tea at Princeton where Montgomery was a visitor.
[excerpt from K. Sabbagh's Dr.
Riemann's Zeros (Atlantic, 2002), on this incident]
The fact that the graph of g descends rapidly as we approach zero means
that the expection values for the numbers of zeta zeros in regions close to a given
zero will rapidly dwindle as we approach zero. It is often said that this suggests
a 'repulsion' between zeros, unlike Poisson (purely random) spacing statistics.
Note that we can relate the Montgomery pairwise correlation conjecture
to the MontgomeryOdlyzko Law discussed above. We see that
since the expectation value of the number of eigenvalues (or zeta zeros) in a
particular interval 'above' a given eigenvalue (or zeta zero) will equal
the sum of probabilities of finding the first, second, third,... eigenvalue
(or zeta zero) in this interval.
F.W.K. Firk and S.J. Miller, "Nuclei, primes and the random matrix connection" (preprint 09/2009)
[abstract:] "In this article, we discuss the remarkable connection between two very different fields, number theory and nuclear physics. We describe the essential aspects of these fields, the quantities studied, and how insights in one have been fruitfully applied in the other. The exciting branch of modern mathematics, random matrix theory, provides the connection between the two fields. We assume no detailed knowledge of number theory, nuclear physics, or random matrix theory; all that is required is some familiarity with linear algebra and probability theory, as well as some results from complex analysis. Our goal is to provide the inquisitive reader with a sound overview of the subjects, placing them in their historical context in a way that is not traditionally given in the popular and technical surveys."
O. Barrett, P. Burkhardt, J. DeWitt, R. Dorward and S.J. Miller, "OneLevel density for holomorphic cusp forms of arbitrary level" (preprint 04/2016)
"In 2000 Iwaniec, Luo, and Sarnak proved for certain families of $L$functions associated to holomorphic newforms of squarefree level that, under the Generalized Riemann Hypothesis, as the conductors tend to infinity the onelevel density of their zeros matches the onelevel density of eigenvalues of large random matrices from certain classical compact groups in the appropriate scaling limit. We remove the squarefree restriction by obtaining a trace formula for arbitrary level by using a basis developed by Blomer and Milicevic, which is of use for other problems as well."
Note: a possible connection between g(r)
and Fibonacci numbers?
[more references can be found in Finch's
notes]
H. Montgomery, "The pair correlation of zeros of the zeta function",
Analytic Number Theory (Proceedings of Symposia in Pure Mathematics
24) (1973) 181193.
J.P. Keating
and
N.C. Snaith,
"Random matrix theory and zeta(1/2 + it)",
Communications in Mathematical Physics 214 (2000) 5789.
This paper is based on part of Nina Snaith's
doctoral research. Her Ph.D. thesis is available here in
PostScript format  the initial chapters serve as an excellent introduction to this whole
area of study.
J.P. Keating and N.C. Snaith,
"Random matrix theory and Lfunctions at
s = 1/2", Communications in Mathematical Physics 214
(2000) 91110.
E. Strahov,
"Moments of characteristic
polynomials enumerate tworowed lexicographic arrays"
[Abstract] "A combinatorial interpretation is provided for the moments of
characteristic polynomials of random unitary matrices. This leads to a rather unexpected
consequence of the Keating and Snaith conjecture: the moments of
$\zeta(1/2+it)$ turn out to be connected with some increasing subsequences
problems (such as the last passage percolation problem).
E. Bogomolny and J.P. Keating, "Random matrix theory and the Riemann zeros
I: three and fourpoint correlations", Nonlinearity, 8 (1995)
1115–1131.
E. Bogomolny and J.P. Keating, "A method for calculating spectral statistics based on randommatrix universality with an application to the threepoint correlations of the Riemann zeros", Journal of Physics A 46 (2013) 305203
[abstract:] "We illustrate a general method for calculating spectral statistics that combines the universal (random matrix theory limit) and the nonuniversal (traceformularelated) contributions by giving a heuristic derivation of the threepoint correlation function for the zeros of the Riemann zeta function. The main idea is to construct a generalized Hermitian random matrix ensemble whose mean eigenvalue density coincides with a large but finite portion of the actual density of the spectrum or the Riemann zeros. Averaging the random matrix result over remaining oscillatory terms related, in the case of the zeta function, to small primes leads to a formula for the threepoint correlation function that is in agreement with results from other heuristic methods. This provides support for these different methods. The advantage of the approach we set out here is that it incorporates the determinental structure of the random matrix limit."
E. Bogomolny and J.P. Keating, "Twopoint correlation function for Dirichlet $L$ functions", Journal of Physics A 46 (2013) 095202
[abstract:] "The twopoint correlation function for the zeros of Dirichlet $L$functions at a height $E$ on the critical line is calculated heuristically using a generalization of the Hardy–Littlewood conjecture for pairs of primes in arithmetic progression. The result matches the conjectured randommatrix form in the limit as $E\rightarrow\infty$ and, importantly, includes finite$E$ corrections. These finite$E$ corrections differ from those in the case of the Riemann zetafunction, obtained in (1996 Phys. Rev. Lett. 77 1472), by certain finite products of primes which divide the modulus of the primitive character used to construct the $L$function in question."
Y.Y. Atas, E. Bogomolny, O. Giraud, P. Vivo and E. Vivo, "Joint probability densities of level spacing ratios in random matrices" (preprint 05/2013)
[abstract:] "We calculate analytically, for finitesize matrices, joint probability densities of ratios of level spacings in ensembles of random matrices characterized by their associated confining potential. We focus on the ratios of two spacings between three consecutive real eigenvalues, as well as certain generalizations such as the overlapping ratios. The resulting formulas are further analyzed in detail in two specific cases: the betaHermite and the betaLaguerre cases, for which we offer explicit calculations for small $N$. The analytical results are in excellent agreement with numerical simulations of usual random matrix ensembles, and with the level statistics of a quantum manybody lattice model and zeros of the Riemann zeta function."
O. Bohigas, P. Leboeuf, M. Sanchez, "On the distribution of the total energy of a system on
noninteracting fermions: random matrix and semiclassical estimates"
"A semiclassical formula describing, as a function of n, a nonuniversal
behavior of the variance of the total energy starting at a critical number
of particles is...obtained. It is illustrated with the particular case
of single particle energies given by the imaginary part of the zeros of
the Riemann zeta function on the critical line."
O. Bohigas, P. Leboeuf,
and M.J. Sanchez,
"Spectral spacing correlations for chaotic and disordered systems"
"New aspects of spectral fluctuations of (quantum) chaotic and
diffusive systems are considered, namely autocorrelations of the
spacing between consecutive levels or spacing autocovariances. They
can be viewed as a discretized two point correlation function. Their
behavior results from two different contributions. One corresponds
to (universal) random matrix eigenvalue fluctuations, the other to
diffusive or chaotic characteristics of the corresponding classical
motion. A closed formula expressing spacing autocovariances in terms
of classical dynamical zeta functions, including the PerronFrobenius
operator, is derived. It leads to a simple interpretation in terms of
classical resonances. The theory is applied to zeros of the Riemann
zeta function. A striking correspondence between the associated
classical dynamical zeta functions and the Riemann zeta itself is
found. This induces a resurgence phenomenon where the lowest Riemann
zeros appear replicated an infinite number of times as resonances and
subresonances in the spacing autocovariances. The theoretical results
are confirmed by existing "data". The present work further extends
the already well known semiclassical interpretation of properties of
Riemann zeros."
P. Leboeuf, A. G. Monastra and O. Bohigas, "The
Riemannium", Regular and Chaotic Dynamics 6 (2001) 205210.
[abstract:] "The properties of a fictitious, fermionic, manybody system based on the complex zeros of the Riemann
zeta function are studied. The imaginary part of the zeros are interpreted as meanfield singleparticle energies, and one fills
them up to a Fermi energy E_{F}. The distribution of the total energy is shown to be nonGaussian,
asymmetric, and independent of E_{F} in the limit E_{F} > infinity. The moments of the
limit distribution are computed analytically. The autocorrelation function, the finite energy corrections, and a comparison with
random matrix theory are also discussed."
P. Leboeuf and A.G. Monastra,
"Quantum thermodynamic
fluctuations of a chaotic Fermigas model"
[abstract:] "We investigate the thermodynamics of a Fermi gas whose singleparticle energy
levels are given by the complex zeros of the Riemann zeta function. This is a model for a
gas, and in particular for an atomic nucleus, with an underlying fully chaotic classical
dynamics. The probability distributions of the quantum fluctuations of the grand potential
and entropy of the gas are computed as a function of temperature and compared, with good
agreement, with general predictions obtained from random matrix theory and periodic orbit
theory (based on prime numbers). In each case the universal and nonuniversal regimes are
identified."
O. Bohigas and M.J. Giannoni, "Chaotic Motions and Random Matrix Theories",
Lecture Notes in Physics 209 (SpringerVerlag, 1984) 199.
E. Bogomolny, O. Bohigas, P. Leboeuf, A. G. Monastra, "On the spacing distribution
of the Riemann zeros: corrections to the asymptotic result" (preprint 02/2006)
[abstract:] "It has been conjectured that the statistical properties of zeros of the Riemann zeta
function near $z = 1/2 + \ui E$ tend, as $E \to \infty$, to the distribution of eigenvalues of large
random matrices from the Unitary Ensemble. At finite $E$ numerical results show that the nearestneighbour
spacing distribution presents deviations with respect to the conjectured asymptotic form. We give here
arguments indicating that to leading order these deviations are the same as those of unitary random matrices
of finite dimension $N_{\rm eff}=\log(E/2\pi)/\sqrt{12 \Lambda}$, where $\Lambda=1.57314 ...$ is a well
defined constant."
P. Forrester and A. Odlyzko,
"A nonlinear equation and its application to nearest neighbor spacings for
zeros of the zeta function and eigenvalues of random matrices"
[abstract:] "A nonlinear equation generalizing the $\omega$ form of the Painleve V
equation is used to compute the probability density function for the
distance from an eigenvalue of a matrix from the GUE ensemble to the
eigenvalue nearest to it. (The classical results concern distribution
of the distances between consecutive eigenvalues.) Comparisons are made
with the corresponding distribution for zeros of the Riemann zeta
function, which are conjectured to behave like eigenvalues of large
random GUE matrices."
P.J. Forrester, "Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble" (preprint 06/2013)
[abstract:] "It is known that the bulk scaling limit of the real eigenvalues for the real Ginibre ensemble is equal in distribution to the rescaled $t \to \infty$ limit of the annihilation process $A + A \to \emptyset$. Furthermore, deleting each particle at random in the rescaled $t \to \infty$ limit of the coalescence process $A + A \to A$, a process equal in distribution to the annihilation process results. We use these interrelationships to deduce from the existing literature the asymptotic small and large distance form of the gap probability for the real Ginibre ensemble. In particular, the leading form of the latter is shown to be equal to $\exp((\zeta(3/2)/(2 \sqrt{2 \pi}))s)$, where $s$ denotes the gap size and $\zeta(z)$ denotes the Riemann zeta function. A determinant formula is derived for the gap probability in the finite $N$ case, and this is used to illustrate the asymptotic formulas against numerical computations."
P.J. Forrester and A. Mays, "Finite size corrections in random matrix theory and Odlyzko's data set for the Riemann zeros" (preprint 07/2015)
[abstract:] "Odlyzko has computed a data set listing more than 109 successive Riemann zeros, starting at a zero number beyond 1023. The data set relates to random matrix theory since, according to the Montgomery—Odlyzko law, the statistical properties of the large Riemann zeros agree with the statistical properties of the eigenvalues of large random Hermitian matrices. Moreover, Keating and Snaith, and then Bogomolny and collaborators, have used $N\times N$ random unitary matrices to analyse deviations from this law. We contribute to this line of study in two ways. First, we point out that a natural process to apply to the data set is to thin it by deleting each member independently with some specified probability, and we proceed to compute empirical twopoint correlation functions and nearest neighbour spacings in this setting. Second, we show how to characterise the order $1/N^2$ correction term to the spacing distribution for random unitary matrices in terms of a second order differential equation with coefficients that are Painlevé transcendents, and where the thinning parameter appears only in the boundary condition. This equation can be solved numerically using a power series method. Comparison with the Riemann zero data shows accurate agreement."
F. Bornemann, P.J. Forrester and A. Mays, "Finite size effects for spacing distributions in random matrix theory: Circular ensembles and Riemann zeros" (preprint 08/2016)
[abstract:] "According to Dyson's three fold way, from the viewpoint of global time reversal symmetry there are three circular ensembles of unitary random matrices relevant to the study of chaotic spectra in quantum mechanics. These are the circular orthogonal, unitary and symplectic ensembles, denoted COE, CUE and CSE respectively. For each of these three ensembles and their thinned versions, whereby each eigenvalue is deleted independently with probability $1  \xi$, we take up the problem of calculating the first two terms in the scaled large $N$ expansion of the spacing distributions. It is well known that the leading term admits a characterisation in terms of both Fredholm determinants and Painlevé transcendents. We show that modifications of these characterisations also remain valid for the next to leading term, and that they provide schemes for high precision numerical computations. In the case of the CUE there is an application to the analysis of Odlyzko's data set for the Riemann zeros, and in that case some further statistics are similarly analysed."
N. Katz and P. Sarnak,
Random Matrices, Frobenius Eigenvalues, and Monodromy
(Colloquium Publications. American Mathematical Society, Vol 45, 1998)
[from the book's introduction:] "In a remarkable numerical experiment, Odlyzko
found that the distribution of the (suitably normalized) spacings between successive
zeroes of the Riemann zeta function is (empirically) the same as the socalled GUE
measure, a certain probability measure on R arising in random matrix theory.
His experiment was inspired by work of Montgomery, who determined the pair
correlation distribution between zeroes (in a restricted range), and who noted the
compatibility of what he found with the GUE prediction. Recent results of Rudnick
and Sarnak are also compatible with the belief that the distribution of the spacings
between zeros, not only of the Riemann zeta function, but also of quite general automorphic
Lfunctions over Q, are all given by the GUE measure, or, as we shall say,
all satisfy the MontgomeryOdlyzko Law. Unfortunately, proving this seems well beyond
the range of existing techniques, and we have no results to offer in this direction.
However, it is a long established principle that problems which seem inaccessible in
the number field case often have finite field analogues which are accessible. In this
book we establish the MontgomeryOdlyzko Law for wide classes of zeta and Lfunctions
over finite fields."
[from a book review:] "Mathematicians from Princeton University focus on the MontgomeryOdlyzko
law, the deep relation between the spacings between zeros of zeta and
Lfunctions and spacings between eigenvalues of random elements of
large compact classical groups. Finds the law to hold for wide classes of
zeta and Lfunctions over finite fields. Of interest to research
mathematicians and graduate students studying such areas as varieties over
finite and local fields, zetafunctions, limit theorems, and the structure of
families."
Z. Rudnick, "Zeta functions in arithmetic and their spectral
statistics", Proceedings of a special semester at the Institut Poincaré, 1996.
[from introduction:] "The Riemann zeta function $\zeta(s)$ serves as an important
model in many investigations into the theory of Quantum Chaos. My aims in these
lectures, which are directed at physicists, are to explain some of the basic properties
of $\zeta(s)$ used by number theorists, and discuss the spectral statistics of their zeros
in connection with Random Matrix Theory."
Z. Rudnick
and P. Sarnak,
"Zeros of principal Lfunctions and
random matrix theory", Duke Mathematics Journal 81 (1996)
269322 (special volume in honour of J. Nash).
E. Brezin, S. Hikami,
"Characteristic polynomials of random matrices"
"Number theorists have studied extensively the connections between the
distribution of zeros of the Riemann zetafunction, and of some generalizations,
with the statistics of the eigenvalues of large random matrices. It is
interesting to compare the average moments of these functions in an interval
to their counterpart in random matrices, which are the expectation values
of the characteristic polynomials of the matrix. It turns out that these
expectation values are quite interesting."
E. Brezin, S. Hikami,
"Logarithmic moments of characteristic polynomials of random matrices"
"In a recent article we have discussed the connections between averages
of powers of Riemann's zetafunction on the critical line, and averages
of characteristic polynomials of random matrices. The result for random
matrices was shown to be universal, i.e. independent of the specific probability
distribution, and the results were derived for arbitrary moments."
J.B. Conrey, M.O. Rubinstein, N.C. Snaith, "Moments of the
derivative of the Riemann zetafunction and of characteristic polynomials" (AIM report no. 2005  20)
[abstract:] "We investigate the moments of the derivative, on the unit circle, of
characteristic polynomials of random unitary matrices and use this to formulate a
conjecture for the moments of the derivative of the Riemann zetafunction on the critical
line. We do the same for the analogue of Hardy's Zfunction, the characteristic polynomial
multiplied by a suitable factor to make it real on the unit circle. Our formulae are
expressed in terms of a determinant of a matrix whose entries involve the IBessel
function and, alternately, by a combinatorial sum."
J.B. Conrey, D.W. Farmer, J.P. Keating, M.O. Rubinstein and N.C. Snaith,
"Integral moments of Lfunctions"
[Abstract:] "We give a new heuristic for all of the main terms in the integral moments of various
families of Lfunctions. The results agree with previous conjectures for the leading order terms.
Our conjectures also have an almost identical form to exact expressions for the corresponding moments
of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments
are defined by the appropriate group averages. This lends support to the idea that arithmetical
Lfunctions have a spectral interpretation, and that their value distributions can be modeled
using Random Matrix Theory. Numerical examples show good agreement with our conjectures."
D. Farmer, F. Mezzadri, N.C. Snaith, "Random
polynomials, random matrices, and Lfunctions, II" (preprint 09/05)
[abstract:] "We show that the Circular Orthogonal Ensemble of random matrices arises naturally
from a family of random polynomials. This sheds light on the appearance of random matrix statistics
in the zeros of the Riemann zetafunction."
[from introduction:] "This paper is motivated by the following questions: do the Riemann zetafunction
and the other Lfunctions of number theory behave differently than random Dirichlet series
with functional equation which just happen to satisfy the Riemann hypothesis? That is, does the Euler product have any effect on the zeros beyond forcing them onto the critical line? Our results
suggest that the answer is 'yes', and the Euler product also has an effect on the local statistics
of the zeros."
F. Mezzadri, "Random matrix theory and
the zeros of zeta'(s)", J. Phys. A 36 (2003), 29452962 (Special Issue on RMT)
[abstract:] "We study the density of the roots of the derivative of the characteristic polynomial
Z(U,z) of an N x N random unitary matrix with distribution given by
Haar measure on the unitary group.
Based on previous random matrix theory models of the Riemann zeta function zeta(s), this is expected to
be an accurate description for the horizontal distribution of the zeros of zeta'(s) to the right of the
critical line. We show that as N > infinity the fraction of roots of Z'(U,z) that lie in the region
1x/(N1) <= z < 1 tends to a limit function. We derive asymptotic expressions for this function in
the limits x > infinity and x > 0 and compare them with numerical experiments."
D.W. Farmer,
"Mean values of the logarithmic derivative of the zeta function and the
GUE hypothesis"
"The GUE Hypothesis, which concerns the
distribution of zeros of the Riemann zetafunction, is used to
evaluate some integrals involving the logarithmic derivative of the
zetafunction. Some connections are shown between the GUE Hypothesis
and other conjectures."
D.W. Farmer, "Modeling families of
Lfunctions" (preprint 11/05)
[abstract:] "We discuss the idea of a 'family of Lfunctions' and describe various methods
which have been used to make predictions about Lfunction families. The methods involve a mixture
of random matrix theory and heuristics from number theory. Particular attention is paid to families of
elliptic curve Lfunctions. We describe two random matrix models for elliptic curve families:
the Independent Model and the Interaction Model."
O. Costin, J.L. Lebowitz,
"Gaussian fluctuations in random matrices", Physical Review
Letters, 75 (1995) 69.
"Let N(L) be the number of eigenvalues, in an interval of length L,
of a matrix chosen at random from the Gaussian Orthogonal, Unitary or
Symplectic ensembles of {\cal N} by {\cal N} matrices, in the limit
{\cal N}\rightarrow\infty. We prove that
[N(L)  \langle N(L)\rangle]/\sqrt{\log L} has a Gaussian distribution
when L\rightarrow\infty. This theorem, which requires control of all
the higher moments of the distribution, elucidates numerical and exact
results on chaotic quantum systems and on the statistics of zeros of
the Riemann zeta function."
C. Hughes, J. Keating,
and N. O'Connell,
"Random Matrix Theory and the Derivative of the Riemann Zeta Function",
Proceedings of the Royal Society A456 (2000) 26112627.
"Random matrix theory (RMT) is used to model the asymptotics of the
discrete moments of the derivative of the Riemann zeta function, $\zeta(s)$,
evaluated at the complex zeros $\tfrac{1}{2}+\i\gamma_n$, using the
methods introduced by Keating and Snaith. We also discuss
the probability distribution of $\ln\zeta'(1/2+\i\gamma_n)$, proving
the central limit theorem for the corresponding random matrix
distribution and analysing its large deviations."
C. Hughes,
"Central limit theorems and large deviations for the characteristic
polynomial of a random unitary matrix and the Riemann zeta function" (a
talk given at
the DMV seminar "The Riemann zeta function and random matrix theory",
Oct 2000)
C. Hughes, J. Keating, and N. O'Connell,
"On the
Characteristic Polynomial of a Random Unitary Matrix" (to be
published in Communications in Mathematical Physics)
"We present a range of fluctuation and large deviations results for
the logarithm of the characteristic polynomial $Z$ of a random $N\times
N$ unitary matrix, as $N\to\infty$. First we show that
$\ln Z/\sqrt{\frac{1}{2}\ln N}$, evaluated at a finite set of distinct
points, is asymptotically a collection of i.i.d. complex normal random
variables. This leads to a refinement of a recent central limit theorem
due to Keating and Snaith, and also explains the covariance structure of
the eigenvalue counting function. We also obtain a central limit theorem
for $\ln Z$ in a Sobolev space of generalised functions on the unit
circle. In this limiting regime, lowerorder terms which reflect the
global covariance structure are no longer negligable and feature in the
covariance structure of the limiting Gaussian measure. Large deviations
results for $\ln Z/A$, evaluated at a finite set of distinct points, can
be obtained for $\sqrt{\ln N} \ll A \ll \ln N$. For higherorder
scalings we obtain large deviations results for $\ln Z/A$ evaluated at a
single point. There is a phase transition at $A=\ln N$ (which only
applies to negative deviations of the real part) reflecting a switch
from global to local conspiracy."
G.A. Hiary, M.O. Rubinstein, "Uniform asymptotics of the coefficients of unitary moment polynomials" (preprint 05/2010)
[abstract:] "Keating and Snaith showed that the $2k^{th}$ absolute moment of the characteristic polynomial of a random unitary matrix evaluated on the unit circle is given by a polynomial of degree $k^2$. In this article, uniform asymptotics for the coefficients of that polynomial are derived, and a maximal coefficient is located. Some of the asymptotics are given in explicit form. Numerical data to support these calculations are presented. Some apparent connections between random matrix theory and the Riemann zeta function are discussed."
Y. Fyodorov, "Negative
moments of characteristic polynomials of random matrices: InghamSiegel
integral as an alternative to HubbardStratonovich transformation",
Nuclear Physics B 621 (2002) 643674.
(Abstract) "We reconsider the problem of calculating arbitrary
negative integer moments of the (regularized) characteristic
polynomial for N x N random matrices taken from the Gaussian
Unitary Ensemble (GUE). A very compact and convenient integral
representation is found via the use of a matrix integral close to that
considered by Ingham and Siegel. We find the asymptotic expression
for the discussed moments in the limit of large N. The latter is of
interest because of a conjectured relation to properties of the
Riemann zetafunction zeroes. Our method reveals a striking
similarity between the structure of the negative and positive integer
moments which is usually obscured by the use of the
HubbardStratonovich transformation. This sheds a new light on
"bosonic" versus "fermionic" replica trick and has some implications
for the supersymmetry method. We briefly discuss the case of the
chiral GUE model from that perspective."
Y.V. Fyodorov, G.A. Hiary and J.P. Keating, "Freezing transition, characteristic polynomials of random matrices, and the Riemann zetafunction" (preprint 02/2012)
[abstract:] "We argue that the freezing transition scenario, previously explored in the statistical mechanics of $1/f$noise random energy models, also determines the value distribution of the maximum of the modulus of the characteristic polynomials of large $N \times N$ random unitary (CUE) matrices. We postulate that our results extend to the extreme values taken by the Riemann zetafunction $\zeta(s)$ over sections of the critical line $s=1/2+it$ of constant length and present the results of numerical computations in support. Our main purpose is to draw attention to possible connections between the statistical mechanics of random energy landscapes, random matrix theory, and the theory of the Riemann zeta function."
Yan V. Fyodorov and Jonathan P. Keating, Freezing transitions and extreme values: Random matrix theory, $\zeta(1/2+it)$, and disordered landscapes (preprint 11/2012)
[abstract:] "We argue that the freezing transition scenario, previously conjectured to occur in the statistical mechanics of $1/f$noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials $p_N(\theta)$ of large $N\times N$ random unitary (CUE) matrices; i.e. the extreme value statistics of $p_N(\theta)$ when $N \rightarrow\infty$. In addition, we argue that it leads to multifractallike behaviour in the total length $\mu_N(x)$ of the intervals in which $p_N(\theta)>N^x$, $x>0$, in the same limit. We speculate that our results extend to the large values taken by the Riemann zetafunction \zeta(s) over stretches of the critical line $s=1/2+it$ of given constant length, and present the results of numerical computations of the large values of $\zeta(1/2+it)$. Our main purpose is to draw attention to possible connections between extreme value problems in the statistical mechanics of $1/f$noise random energy models, random matrix theory, and the theory of the Riemann zeta function, and to the potential consequences of freezing in the latter two cases."
Z. Ahmed and S.R. Jain, "A pseudounitary ensemble of
random matrices, PTsymmetry and the Riemann Hypothesis"
(preprint 07/04)
[abstract:] "An ensemble of 2 x 2 pseudoHermitian random matrices is constructed that possesses real eigenvalues with
levelspacing distribution exactly as for the Gaussian Unitary Ensemble found by Wigner. By a reinterpretation of Connes'
spectral interpretation of the zeros of the Riemann zeta function, we propose to enlarge the scope of search of the Hamiltonian
connected with the celebrated Riemann Hypothesis by suggesting that the Hamiltonian could also be PTsymmetric
(or pseudoHermitian)."
Z. Ahmed, "Gaussianrandom Ensembles of PseudoHermitian
Matrices", Invited Talk Delivered in 2^{nd} International Workshop on 'PseudoHermitian Hanmiltonians in Physics',
Prague, June 1416, 2004
[abstract:] "Attention has been brought to the possibility that statistical fluctuation properties of several complex spectra, or,
wellknown number sequences may display strong signatures that the Hamiltonian yielding them as eigenvalues is PTsymmetric
(PseudoHermitian). We find that the random matrix theory of pseudoHermitian Hamiltonians gives rise to new universalities of
levelspacing distributions other than those of GOE, GUE and GSE of Wigner and Dyson. We call the new proposals as Gaussian
PseudoOrthogonal Ensemble and Gaussian PseudoUnitary Ensemble. We are also led to speculate that the enigmatic
Riemannzeros would rather correspond to some PTsymmetric (pseudoHermitian) Hamiltonian. "
R. Chhaibi, J. Najnudel and A. Nikeghbali, "The Circular Unitary Ensemble and the Riemann zeta function: The microscopic landscape" (preprint 10/2014)
[abstract:] "We show in this paper that after proper scalings, the characteristic polynomial of a random unitary matrix converges almost surely to a random analytic function whose zeros, which are on the real line, form a determinantal point process with sine kernel. Our scaling is performed at the socalled "microscopic" level, that is we consider the characteristic polynomial at points which are of order $1/n$ distant. We draw several consequences from our result. On the random matrix theory side, we obtain the limiting distribution for ratios of characteristic polynomials where the points are evaluated at points of the form $\exp(2i\pi\alpha/n)$. We also give an explicit expression for the (dependence) relation between two different values of the characteristic polynomial on the microscopic scale. On the number theory side, inspired by the Keating–Snaith philosophy, we conjecture some new limit theorems for the Riemann zeta function at the stochastic process level as well as some alternative approach to the conjecture by Goldston, Montgomery and Gonek for the moments of the logarithmic derivative of the Riemann zeta function. We prove our main random matrix theory result in the framework of virtual isometries to circumvent the fact that the rescaled characteristic polynomial does not even have a moment of order one, hence making the classical techniques of random matrix theory difficult to apply."
G.B. Arous, P. Bourgade, "Extreme gaps between eigenvalues of random matrices" (preprint 10/2010)
[abstract:] "This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haardistributed unitary matrices and matrices from the Gaussian Unitary Ensemble. In particular, the $k$th smallest gap, normalized by a factor $n^{4/3}$, has a limiting density proportional to $x^{3k1}e^{x^3}$. Concerning the largest gaps, normalized by $n/\sqrt{\log n}$, they converge in $\L^p$ to a constant for all $p>0$. These results are compared with the extreme gaps between zeros of the Riemann zeta function."
A. Edelman, P.O. Persson, "Numerical methods for
eigenvalue distributions of random matrices (preprint 01/05)
[abstract:] "We present efficient numerical techniques for calculation of eigenvalue distributions of
random matrices in the betaensembles. We compute histograms using direct simulations on very large matrices,
by using tridiagonal matrices with appropriate simplifications. The distributions are also obtained by
numerical solution of the Painleve II and V equations with high accuracy. For the spacings we show a technique
based on the Prolate matrix and Richardson extrapolation, and we compare the distributions with the zeros of
the Riemann zeta function."
P. Diaconis and M. Coram,
"New
Tests of the Correspondence Between Unitary Eigenvalues and the zeros
of Riemann's Zeta Function"
[PostScript]
T. Kriecherbauer, J. Marklof and A. Soshnikov,
"Random matrices and quantum chaos"
(brief introductory article, including a description of how this
relates to the Riemann Hypothesis)
J.F. Burnol, speculative
excerpt from "On Fourier and zeta(s)"
relating the GUE hypothesis to his earlier work on physical interpretations
of number theoretical phenomena.
E. Wigner, "Random matrices in physics", SIAM Review 9
(1967) 1123.
A.V. Andreev, O. Agam, B.D. Simons, B.L. Altschuler, "Quantum chaos,
irreversible classical dynamics, and random matrix theory", Physical Review
Letters 76 (1996) 3497
P. Deift, "Universality for mathematical and physical
systems" (preprint 03/2006)
[abstract:] "All physical systems in equilibrium obey the laws of thermodynamics. In other words, whatever the precise
nature of the interaction between the atoms and molecules at the microscopic level, at the macroscopic level, physical
systems exhibit universal behavior in the sense that they are all governed by the same laws and formulae of thermodynamics.
In this paper we describe some recent history of universality ideas in physics starting with Wigner's model for the
scattering of neutrons off large nuclei and show how these ideas have led mathematicians to investigate universal behavior
for a variety of mathematical systems. This is true not only for systems which have a physical origin, but also for systems
which arise in a purely mathematical context such as the Riemann hypothesis, and a version of the card game solitaire
called patience sorting."
J.B. Conrey, D.W. Farmer, J.P. Keating, M.O. Rubinstein, N.C. Snaith,
"Lower order terms in the full moment conjecture for the Riemann zeta
function" (preprint 12/2006)
[abstract:] "We describe an algorithm for obtaining explicit expressions for lower terms for the conjectured full asymptotics of the
moments of the Riemann zeta function, and give two distinct methods for obtaining numerical values of these coefficients. We also provide
some numerical evidence in favour of the conjecture."
N.Kobayashi, M. Izumi, M. Katori, "Maximum distributions of noncolliding Bessel bridges" (preprint 08/2008)
[abstract:] "The onedimensional Brownian motion starting from the origin at time $t=0$, conditioned to return to the origin at time $t=1$ and to stay positive during time
interval $0 < t < 1$, is called the Bessel bridge with duration 1. We consider the $N$particle system of such Bessel bridges conditioned never to collide with each other in
$0 < t < 1$, which is the continuum limit of the vicious walk model in watermelon configuration with a wall. Distributions of maximumvalues of paths attained in the time interval
$t \in (0,1)$ are studied to characterize the statistics of random patterns of the repulsive paths on the spatiotemporal plane. For the outermost path, the distribution function of
maximum value is exactly determined for general $N$. We show that the present $N$path system of noncolliding Bessel bridges is realized as the positiveeigenvalue process
of the $2N \times 2N$ matrixvalued Brownian bridge in the symmetry class C. Using this fact computer simulations are performed and numerical results on the $N$dependence
of the maximumvalue distributions of the inner paths are reported. The present work demonstrates that the extremevalue problems of noncolliding paths are related with the
random matrix theory, representation theory of symmetry, and the number theory."
S. Torquato, A. Scardicchio, C.E. Zachary, "Point processes in arbitrary dimension from
fermionic gases, random matrix theory, and number theory" (preprint 09/2008)
[abstract:] "It is well known that one can map certain properties of random matrices, fermionic gases, and zeros of the Riemann zeta function to a unique point process
on the real line. Here we analytically provide exact generalizations of such a point process in $d$dimensional Euclidean space for any $d$, which are special cases of
determinantal processes. In particular, we obtain the $n$particle correlation functions for any n, which completely specify the point processes. We also demonstrate that
spinpolarized fermionic systems have these same $n$particle correlation functions in each dimension. The point processes for any $d$ are shown to be hyperuniform. The
latter result implies that the pair correlation function tends to unity for large pair distances with a decay rate that is controlled by the power law $r^{(d+1)}$. We graphically
display one and twodimensional realizations of the point processes in order to vividly reveal their "repulsive" nature. Indeed, we show that the point processes can be
characterized by an effective "hardcore" diameter that grows like the square root of $d$. The nearestneighbor distribution functions for these point processes are also
evaluated and rigorously bounded. Among other results, this analysis reveals that the probability of finding a large spherical cavity of radius $r$ in dimension $d$ behaves
like a Poisson point process but in dimension $d+1$ for large r and finite $d$. We also show that as $d$ increases, the point process behaves effectively like a sphere packing
with a coverage fraction of space that is no denser than $1/2^d$."
S. Tanaka, "Distribution of the Riemann zeros represented by the Fermi gas" (preprint 10/2010)
[abstract:] "The multiparticle density matrices for degenerate, ideal Fermi gas system in any dimension are calculated. The results are expressed as a determinant form, in which a correlation kernel plays a vital role. Interestingly, the correlation structure of onedimensional Fermi gas system is essentially equivalent to that observed for the eigenvalue distribution of random unitary matrices, and thus to that conjectured for the distribution of the nontrivial zeros of the Riemann zeta function. Implications of the present findings are discussed briefly. "
S. Ali Altug, Sandro Bettin, Ian Petrow, Rishikesh and Ian Whitehead, "A recursion formula for moments of derivatives of random matrix polynomials" (preprint 12/2012)
[abstract:] "We give asymptotic formulae for random matrix averages of derivatives of characteristic polynomials over the groups $USp(2N)$, $SO(2N)$ and $O^(2N)$. These averages are used to predict the asymptotic formulae for moments of derivatives of $L$functions which arise in number theory. Each formula gives the leading constant of the asymptotic in terms of determinants of hypergeometric functions. We find a differential recurrence relation between these determinants which allows the rapid computation of the $(k+1)$st constant in terms of the $k$th and $(k1)$st. This recurrence is reminiscent of a Toda lattice equation arising in the theory of $\tau$functions associated with Painlevé differential equations."
C. Pineda and T. Prosen, "Nonuniversal level statistics in a chaotic quantum spin chain", Phys. Rev. E 76 (2007) 061127
[abstract:] "We study the level statistics of an interacting multiqubit system, namely the kicked Ising spin chain, in the regime of quantum chaos. Long range quasienergy level statistics show effects analogous to the ones observed in semiclassical systems due to the presence of classical periodic orbits, while short range level statistics display perfect statistical agreement with random matrix theory. Even though our system possesses no classical limit, our result suggest existence of an important nonuniversal system specific behavior at short time scale, which clearly goes beyond finite size effects in random matrix theory."
[author comment:] "[We attempt] to calculate the dimension of a Hilbert space associated with rotationally invariant systems of $n$ spins. The dimension was given in terms of the Möbius function."
J. Sakhr and J.M. Nieminen, "Local boxcounting dimensions of discrete quantum eigenvalue spectra: Analytical connection to quantum spectral statistics" (preprint 11/2017)
[abstract:] "Two decades ago, Wang and Ong [Phys. Rev. A 55, 1522 (1997)] hypothesized that the local boxcounting dimension of a discrete quantum spectrum should depend exclusively on the nearestneighbor spacing distribution (NNSD) of the spectrum. In this paper, we validate their hypothesis by deriving an explicit formula for the local boxcounting dimension of a countablyinfinite discrete quantum spectrum. This formula expresses the local boxcounting dimension of a spectrum in terms of single and double integrals of the NNSD of the spectrum. As applications, we derive an analytical formula for Poisson spectra and closedform approximations to the local boxcounting dimension for spectra having Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE), and Gaussian symplectic ensemble (GSE) spacing statistics. In the Poisson and GOE cases, we compare our theoretical formulas with the published numerical data of Wang and Ong and observe excellent agreement between their data and our theory. We also study numerically the local boxcounting dimensions of the Riemann zeta function zeros and the alternate levels of GOE spectra, which are often used as numerical models of spectra possessing GUE and GSE spacing statistics, respectively. In each case, the corresponding theoretical formula is found to accurately describe the numericallycomputed local boxcounting dimension."
A. Chattopadhyay, P. Dutta, S. Dutta and D. Ghoshal, "Matrix model for Riemann zeta via its local factors" (preprint 07/2018)
[abstract:] "We propose the construction of an ensemble of unitary random matrices (UMM) for the Riemann zeta function. Our approach to this problem is 'piecemeal', in the sense that we consider each factor in the Euler product representation of the zeta function to first construct a UMM for each prime $p$. We are able to use its phase space description to write the partition function as the trace of an operator that acts on a subspace of squareintegrable functions on the $p$adic line. This suggests a Berry–Keating type Hamiltonian. We combine the data from all primes to propose a Hamiltonian and a matrix model for the Riemann zeta function."
S.C.L. Srivastava, A. Lakshminarayan, S. Tomsovic and A. Bäcker, "Ordered level spacing probability densities" (preprint 10/2018)
[abstract:] "Spectral statistics of quantum systems have been studied in detail using the nearest neighbour level spacings, which for generic chaotic systems follows random matrix theory predictions. In this work, the probability density of the closest neighbour and farther neighbour spacings from a given level are introduced. Analytical predictions are derived using a $3\times 3$ matrix model. The closest neighbour density is generalized to the $k$th closest neighbour spacing density, which allows for investigating longrange correlations. For larger $k$ the probability density of $k$th closest neighbour spacings is well described by a Gaussian. Using these $k$th closest neighbour spacings we propose the ratio of the closest neighbour to the second closest neighbour as an alternative to the ratio of successive spacings. For a Poissonian spectrum the density of the ratio is flat, whereas for the three Gaussian ensembles repulsion at small values is found. The ordered spacing statistics and their ratio are numerically studied for the integrable circle billiard, the chaotic cardioid billiard, the standard map and the zeroes of the Riemann zeta function. Very good agreement with the predictions is found."
Abstracts from MSRI conference "Random Matrices and Their Applications:
Quantum Chaos, GUE Conjecture for Zeros of Zeta Functions, Combinatorics,
and All That" (June 711, 1999)
MSRI archive  streaming video of numerous lectures on random matrix theory:
Random
Matrix, Statistical Mechanics, and Integrable Systems Workshop, February 2226, 1999
Random
Matrices and their Applications, June 711, 1999
Random Matrices
Conference, MIT, 12 August 2001.
Zeta Functions,
Random Matrices and Quantum Chaos Workshop,
Bristol, UK, 1314 September, 2001.
P.E. Cartier, B. Julia, P. Moussa and P. Vanhove (eds.), Frontiers in Number Theory,
Physics, and Geometry: On Random Matrices, Zeta Functions, and Dynamical Systems (Springer, due March 2006)
[publisher's description:] "This book presents pedagogical contributions on selected topics relating Number Theory, Theoretical
Physics and Geometry. The parts are composed of long selfcontained pedagogical lectures followed by shorter contributions
on specific subjects organized by theme. Most courses and short contributions go up to the recent developments in the fields;
some of them follow their author's original viewpoints. There are contributions on Random Matrix Theory, Quantum Chaos,
Noncommutative Geometry, Zeta functions, and Dynamical Systems. The chapters of this book are extended versions of
lectures given at a meeting entitled Number Theory, Physics and Geometry,
held at Les Houches in March 2003."
"Recent
Perspectives in Random Matrix Theory and Number Theory", Isaac Newton Institute of Mathematical
Sciences, Cambridge, UK, 29 March  8 April 2004
This was linked with the Isaac Newton Institute programme:
"Random Matrix Approaches in Number
Theory", 26 January  16 July 2004
Workshop on Number Theory
and Random Matrix Theory, June 13, 2005, Waterloo, Canada
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