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The loss surfaces of deep neural networks have been the subject of several studies, theoretical and experimental, over the last few years. One strand of work considers the complexity, in the sense of local optima, of high dimensional random functions with the aim of informing how local optimisation methods may perform in such complicated settings. Prior work of Choromanska et al (2015) established a direct link between the training loss surfaces of deep multi-layer perceptron networks and spherical multi-spin glass models under some very strong assumptions on the network and its data. In this work, we test the validity of this approach by removing the undesirable restriction to ReLU activation functions. In doing so, we chart a new path through the spin glass complexity calculations using supersymmetric methods in Random Matrix Theory which may prove useful in other contexts. Our results shed new light on both the strengths and the weaknesses of spin glass models in this context., 50 pages, 11 figures; references added for Kac-Rice reduction to RMT method; updates following JSTAT review and publication

Whereas much work in the mathematical literature on quantum chaos has focused on phenomena such as quantum ergodicity and scarring, relatively little is known at the rigorous level about the existence of eigenfunctions whose morphology is more complex. Quantum systems whose dynamics is intermediate between certain regimes - for example, at the transition between Anderson localized and delocalized eigenfunctions, or in systems whose classical dynamics is intermediate between integrability and chaos - have been conjectured in the physics literature to have eigenfunctions exhibiting multifractal, self-similar structure. To-date, no rigorous mathematical results have been obtained about systems of this kind in the context of quantum chaos. We give here the first rigorous proof of the existence of multifractal eigenfunctions for a widely studied class of intermediate quantum systems. Specifically, we derive an analytical formula for the Renyi entropy associated with the eigenfunctions of arithmetic Seba billiards, in the semiclassical limit, as the associated eigenvalues tend to infinity. We also prove multifractality of the ground state for more general, non-arithmetic billiards and show that the fractal exponent in this regime satisfies a symmetry relation, similar to the one predicted in the physics literature, by establishing a connection with the functional equation for Epstein's zeta function., 28 pages, to appear in Comm. Math. Phys

We consider powers of the absolute value of the characteristic polynomial of Haar distributed random orthogonal or symplectic matrices, as well as powers of the exponential of its argument, as a random measure on the unit circle minus small neighborhoods around $\pm 1$. We show that for small enough powers and under suitable normalization, as the matrix size goes to infinity, these random measures converge in distribution to a Gaussian multiplicative chaos measure. Our result is analogous to one on unitary matrices previously established by Christian Webb in [31]. We thus complete the connection between the classical compact groups and Gaussian multiplicative chaos. To prove this we establish appropriate asymptotic formulae for Toeplitz and Toeplitz+Hankel determinants with merging singularities. Using a recent formula communicated to us by Claeys et al., we are able to extend our result to the whole of the unit circle., 63 pages, 7 figues

Representation theory and the theory of symmetric functions have played a central role in Random Matrix Theory in the computation of quantities such as joint moments of traces and joint moments of characteristic polynomials of matrices drawn from the Circular Unitary Ensemble and other Circular Ensembles related to the classical compact groups. The reason is that they enable the derivation of exact formulae, which then provide a route to calculating the large-matrix asymptotics of these quantities. We develop a parallel theory for the Gaussian Unitary Ensemble of random matrices, and other related unitary invariant matrix ensembles. This allows us to write down exact formulae in these cases for the joint moments of the traces and the joint moments of the characteristic polynomials in terms of appropriately defined symmetric functions. As an example of an application, for the joint moments of the traces we derive explicit asymptotic formulae for the rate of convergence of the moments of polynomial functions of GUE matrices to those of a standard normal distribution when the matrix size tends to infinity., Minor corrections. To appear in Journal of Mathematical Physics. 44 pages

We compute the variances of sums in arithmetic progressions of generalized [Formula: see text]-divisor functions related to certain [Formula: see text]-functions in [Formula: see text], in the limit as [Formula: see text]. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when [Formula: see text], in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual [Formula: see text]-divisor function, when the [Formula: see text]-function in question has degree one. They illustrate the role played by the degree of the [Formula: see text]-functions; in particular, we find qualitatively new behavior when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over [Formula: see text], and we illustrate them by examining in some detail the generalized [Formula: see text]-divisor functions associated with the Legendre curve.

The auto-correlations of arithmetic functions, such as the von Mangoldt function, the M\"obius function and the divisor function, are the subject of classical problems in analytic number theory. The function field analogues of these problems have recently been resolved in the limit of large finite field size $q$. However, in this limit the correlations disappear: the arithmetic functions become uncorrelated. We compute averages of terms of lower order in $q$ which detect correlations. Our results show that there is considerable cancellation in the averaging and have implications for the rate at which correlations disappear when $q \rightarrow\infty$; in particular one cannot expect remainder terms that are of the order of the square-root of the main term in this context., Comment: The paper has been accepted by IMRN

Denoting by $P_N(A,\theta)=\det(I-Ae^{-i\theta})$ the characteristic polynomial on the unit circle in the complex plane of an $N\times N$ random unitary matrix $A$, we calculate the $k$th moment, defined with respect to an average over $A\in U(N)$, of the random variable corresponding to the $2\beta$th moment of $P_N(A,\theta)$ with respect to the uniform measure $\frac{d\theta}{2\pi}$, for all $k, \beta\in\mathbb{N}$ . These moments of moments have played an important role in recent investigations of the extreme value statistics of characteristic polynomials and their connections with log-correlated Gaussian fields. Our approach is based on a new combinatorial representation of the moments using the theory of symmetric functions, and an analysis of a second representation in terms of multiple contour integrals. Our main result is that the moments of moments are polynomials in $N$ of degree $k^2\beta^2-k+1$. This resolves a conjecture of Fyodorov \& Keating~\cite{fyodorov14} concerning the scaling of the moments with $N$ as $N\rightarrow\infty$, for $k,\beta\in\mathbb{N}$. Indeed, it goes further in that we give a method for computing these polynomials explicitly and obtain a general formula for the leading coefficient., Comment: 35 pages, to appear in CMP

In this series of papers we examine the calculation of the $2k$th moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper completes the general study of what we call Type II sums which utilize a circle method framework and a convolution of shifted convolution sums to obtain all of the lower order terms in the asymptotic formula for the mean square along $[T,2T]$ of a Dirichlet polynomial of arbitrary length with divisor functions as coefficients., Comment: Revised version; accepted in PLMS

We establish a representation of the joint moments of the characteristic polynomial of a CUE random matrix and its derivative in terms of a solution of the sigma-Painleve V equation. The derivation involves the analysis of a formula for the joint moments in terms of a determinant of generalised Laguerre polynomials using the Riemann-Hilbert method. We use this connection with the sigma-Painleve V equation to derive explicit formulae for the joint moments and to show that in the large-matrix limit the joint moments are related to a solution of the sigma-Painleve III equation. Using the conformal block expansion of the tau-functions associated with the sigma-Painleve V and the sigma-Painleve III equations leads to general conjectures for the joint moments., 39 pages. To appear in Nonlinearity

Nonlinearity has published articles containing a significant number-theoretic component since the journal was first established. We examine one thread, concerning the statistics of the zeros of the Riemann zeta function. We extend this by establishing a connection between the ratios conjecture for the Riemann zeta-function and a conjecture concerning correlations of convolutions of Möbius and divisor functions. Specifically, we prove that the ratios conjecture and an arithmetic correlations conjecture imply the same result. This provides new support for the ratios conjecture, which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe. Our main theorem generalises a recent calculation pertaining to the special case of two-over-two ratios.

We exploit mappings between quantum and classical systems in order to obtain a class of two-dimensional classical systems with critical properties equivalent to those of the class of one-dimensional quantum systems discussed in a companion paper (J. Hutchinson, J. P. Keating, and F. Mezzadri, arXiv:1503.05732). In particular, we use three approaches: the Trotter-Suzuki mapping; the method of coherent states; and a calculation based on commuting the quantum Hamiltonian with the transfer matrix of a classical system. This enables us to establish universality of certain critical phenomena by extension from the results in our previous article for the classical systems identified., Comment: 36 pages

We examine the calculation of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. Previously, this approach has proved unsuccessful in computing moments beyond the eighth, even heuristically. A careful analysis of the second and fourth moments illustrates the nature of the problem and enables us to identify the terms that are missed in the standard application of these methods.

We argue that thefreezing 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 polynomialspN(θ) of largeN×Nrandom unitary (circular unitary ensemble) matricesUN; i.e. the extreme value statistics ofpN(θ) when. In addition, we argue that it leads to multi-fractal-like behaviour in the total lengthμN(x) of the intervals in which |pN(θ)|>Nx,x>0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta functionζ(s) over stretches of the critical lineof given constant length and present the results of numerical computations of the large values of). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.

We develop a full characterization of abelian quantum statistics on graphs. We explain how the number of anyon phases is related to connectivity. For 2-connected graphs the independence of quantum statistics with respect to the number of particles is proven. For non-planar 3-connected graphs we identify bosons and fermions as the only possible statistics, whereas for planar 3-connected graphs we show that one anyon phase exists. Our approach also yields an alternative proof of the structure theorem for the first homology group of n-particle graph configuration spaces. Finally, we determine the topological gauge potentials for 2-connected graphs., 34 pages, 21 figures

We propose a random matrix model for families of elliptic curve L-functions of finite conductor. A repulsion of the critical zeros of these L-functions away from the center of the critical strip was observed numerically by S. J. Miller in 2006; such behaviour deviates qualitatively from the conjectural limiting distribution of the zeros (for large conductors this distribution is expected to approach the one-level density of eigenvalues of orthogonal matrices after appropriate rescaling).Our purpose here is to provide a random matrix model for Miller's surprising discovery. We consider the family of even quadratic twists of a given elliptic curve. The main ingredient in our model is a calculation of the eigenvalue distribution of random orthogonal matrices whose characteristic polynomials are larger than some given value at the symmetry point in the spectra. We call this sub-ensemble of SO(2N) the excised orthogonal ensemble. The sieving-off of matrices with small values of the characteristic polynomial is akin to the discretization of the central values of L-functions implied by the formula of Waldspurger and Kohnen-Zagier.The cut-off scale appropriate to modeling elliptic curve L-functions is exponentially small relative to the matrix size N. The one-level density of the excised ensemble can be expressed in terms of that of the well-known Jacobi ensemble, enabling the former to be explicitly calculated. It exhibits an exponentially small (on the scale of the mean spacing) hard gap determined by the cut-off value, followed by soft repulsion on a much larger scale. Neither of these features is present in the one-level density of SO(2N). When N tends to infinity we recover the limiting orthogonal behaviour. Our results agree qualitatively with Miller's discrepancy. Choosing the cut-off appropriately gives a model in good quantitative agreement with the number-theoretical data., 38 pages, version 2 (added some plots)

We describe some new families of quasimodes for the Laplacian perturbed by the addition of a potential formally described by a Dirac delta function. As an application we find, under some additional hypotheses on the spectrum, subsequences of eigenfunctions of Seba billiards that localise around a pair of unperturbed eigenfunctions., Comment: 23 pages, 1 figure

We establish an asymptotic formula for the number of k -difference twin primes associated with the Hawkins random sieve, which is a probabilistic model of the Eratosthenes sieve. The formula for k = 1 was obtained by M.C. Wunderlich [A probabilistic setting for prime number theory, Acta Arith. 26 (1974) 59–81]. We here extend this to k ⩾ 2 and generalize it to all l -tuples of Hawkins primes.

We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakonski et al (J. Phys. A, v. 34, 9303-9317 (2001)). As observables we take the L^2 functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance. Specifically, given a one-dimensional, Lebesgue-measure-preserving map of an interval, we consider an increasingly refined sequence of partitions of the interval. To this sequence we associate a sequence of graphs, whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the Perron-Frobenius operator corresponding to the map. We show that, except possibly for subsequences of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs. For a smaller class of observables we also show that the Egorov property, a correspondence between classical and quantum evolution in the semiclassical limit, holds for the quantum graphs in question., 20 pages, 1 figure

We propose a new correlator in one-dimensional quantum spin chains, the $s-$Emptiness Formation Probability ($s-$EFP). This is a natural generalization of the Emptiness Formation Probability (EFP), which is the probability that the first $n$ spins of the chain are all aligned downwards. In the $s-$EFP we let the spins in question be separated by $s$ sites. The usual EFP corresponds to the special case when $s=1$, and taking $s>1$ allows us to quantify non-local correlations. We express the $s-$EFP for the anisotropic XY model in a transverse magnetic field, a system with both critical and non-critical regimes, in terms of a Toeplitz determinant. For the isotropic XY model we find that the magnetic field induces an interesting length scale., 6 pages, 1 figure

Bipartite entanglement in the ground state of a chain of $N$ quantum spins can be quantified either by computing pairwise concurrence or by dividing the chain into two complementary subsystems. In the latter case the smaller subsystem is usually a single spin or a block of adjacent spins and the entanglement differentiates between critical and non-critical regimes. Here we extend this approach by considering a more general setting: our smaller subsystem $S_A$ consists of a {\it comb} of $L$ spins, spaced $p$ sites apart. Our results are thus not restricted to a simple `area law', but contain non-local information, parameterized by the spacing $p$. For the XX model we calculate the von-Neumann entropy analytically when $N\to \infty$ and investigate its dependence on $L$ and $p$. We find that an external magnetic field induces an unexpected length scale for entanglement in this case., 6 pages, 4 figures

We compute the entropy of entanglement in the ground states of a general class of quantum spin-chain Hamiltonians - those that are related to quadratic forms of Fermi operators - between the first N spins and the rest of the system in the limit of infinite total chain length. We show that the entropy can be expressed in terms of averages over the classical compact groups and establish an explicit correspondence between the symmetries of a given Hamiltonian and those characterizing the Haar measure of the associated group. These averages are either Toeplitz determinants or determinants of combinations of Toeplitz and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture are used to compute the leading order asymptotics of the entropy as N --> infinity . This is shown to grow logarithmically with N. The constant of proportionality is determined explicitly, as is the next (constant) term in the asymptotic expansion. The logarithmic growth of the entropy was previously predicted on the basis of numerical computations and conformal-field-theoretic calculations. In these calculations the constant of proportionality was determined in terms of the central charge of the Virasoro algebra. Our results therefore lead to an explicit formula for this charge. We also show that the entropy is related to solutions of ordinary differential equations of Painlev�� type. In some cases these solutions can be evaluated to all orders using recurrence relations., 39 pages, 1 table, no figures. Revised version: minor corrections

We investigate statistical properties of the eigenfunctions of the Schrodinger operator on families of star graphs with incommensurate bond lengths. We show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average. We further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds. We describe how to construct such subsequences explicitly. These constructions are analogous to scars on short unstable periodic orbits., 26 pages, 5 figures

The statistics of the nodal lines and nodal domains of the eigenfunctions of quantum billiards have recently been observed to be fingerprints of the chaoticity of the underlying classical motion by Blum et al. (Phys. Rev. Lett., Vol. 88 (2002), 114101) and by Bogomolny and Schmit (Phys. Rev. Lett., Vol. 88 (2002), 114102). These statistics were shown to be computable from the random wave model of the eigenfunctions. We here study the analogous problem for chaotic maps whose phase space is the two-torus. We show that the distributions of the numbers of nodal points and nodal domains of the eigenvectors of the corresponding quantum maps can be computed straightforwardly and exactly using random matrix theory. We compare the predictions with the results of numerical computations involving quantum perturbed cat maps., 7 pages, 2 figures. Second version: minor corrections

We compute the value distributions of the eigenfunctions and spectral determinant of the Schrodinger operator on families of star graphs. The values of the spectral determinant are shown to have a Cauchy distribution with respect both to averages over bond lengths in the limit as the wavenumber tends to infinity and to averages over wavenumber when the bond lengths are fixed and not rationally related. This is in contrast to the spectral determinants of random matrices, for which the logarithm is known to satisfy a Gaussian limit distribution. The value distribution of the eigenfunctions also differs from the corresponding random matrix result. We argue that the value distributions of the spectral determinant and of the eigenfunctions should coincide with those of Seba-type billiards., 32 pages, 9 figures. Final version incorporating referee's comments. Typos corrected, appendix added

We derive an exact expression for the two-point correlation function for quantum star graphs in the limit as the number of bonds tends to infinity. This turns out to be identical to the corresponding result for certain Seba billiards in the semiclassical limit. Reasons for this are discussed. The formula we derive is also shown to be equivalent to a series expansion for the form factor - the Fourier transform of the two-point correlation function - previously calculated using periodic orbit theory., Comment: 24 pages, 3 figures, submitted to J. Phys. A

According to theorems of Shnirelman and followers, in the semiclassical limit the quantum wavefunctions of classically ergodic systems tend to the microcanonical density on the energy shell. We here develop a semiclassical theory that relates the rate of approach to the decay of certain classical fluctuations. For uniformly hyperbolic systems we find that the variance of the quantum matrix elements is proportional to the variance of the integral of the associated classical operator over trajectory segments of length $T_H$, and inversely proportional to $T_H^2$, where $T_H=h\bar\rho$ is the Heisenberg time, $\bar\rho$ being the mean density of states. Since for these systems the classical variance increases linearly with $T_H$, the variance of the matrix elements decays like $1/T_H$. For non-hyperbolic systems, like Hamiltonians with a mixed phase space and the stadium billiard, our results predict a slower decay due to sticking in marginally unstable regions. Numerical computations supporting these conclusions are presented for the bakers map and the hydrogen atom in a magnetic field., Comment: 11 pages postscript and 4 figures in two files, tar-compressed and uuencoded using uufiles, to appear in Phys Rev E. For related papers, see http://www.icbm.uni-oldenburg.de/icbm/kosy/ag.html

We calculate the first and second moments of L-functions in the family of quadratic twists of a fixed elliptic curve E over F_q[x], asymptotically in the limit as the degree of the twists tends to infinity. We also compute moments involving derivatives of L-functions over quadratic twists, enabling us to deduce lower bounds on the correlations between the analytic ranks of the twists of two distinct curves., 32 pages

We compute the variances of sums in arithmetic progressions of arithmetic functions associated with certain L-functions of degree 2 and higher in F q [t], in the limit as q →∞. This is achieved by establishing appropriate equidistribution results for the associated Frobenius conjugacy classes. The variances are thus related to matrix integrals, which may be evaluated. Our results differ significantly from those that hold in the case of degree-1 L-functions (i.e., situations considered previously using this approach). They correspond to expressions found recently in the number field setting assuming a generalization of the pair correlation conjecture. Our calculations apply, for example, to elliptic curves defined over F q [t].

We present the results of systematic numerical computations relating to the extreme value statistics of the characteristic polynomials of random unitary matrices drawn from the Circular Unitary Ensemble (CUE) of Random Matrix Theory. In particular, we investigate a range of recent conjectures and theoretical results inspired by analogies with the theory of logarithmically-correlated Gaussian random fields. These include phenomena related to the conjectured freezing transition. Our numerical results are consistent with, and therefore support, the previous conjectures and theory. We also go beyond previous investigations in several directions: we provide the first quantitative evidence in support of a correlation between extreme values of the characteristic polynomials and large gaps in the spectrum, we investigate the rate of convergence to the limiting formulae previously considered, and we extend the previous analysis of the CUE to the C$\beta$E which corresponds to allowing the degree of the eigenvalue repulsion to become a parameter., Comment: 23 pages