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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

Using the Ratios Conjecture, we write down precise formulas with lower order terms for the one and the two level densities of zeros of quadratic Dirichlet $L$--functions over function fields. We denote the various terms arising as Type-$0$, Type-I and Type-II contributions. When the support of the Fourier transform of the test function is sufficiently restricted, we rigorously compute the Type-$0$ and Type-I terms and confirm that they match the conjectured answer. When the restrictions on the support are relaxed, our results suggest that Type-II contributions become important in the two level density., Comment: 23 pages

We study the mean square of sums of the kth divisor function $$d_k(n)$$ over short intervals and arithmetic progressions for the rational function field over a finite field of q elements. In the limit as $$q\rightarrow \infty $$ we establish a relationship with a matrix integral over the unitary group. Evaluating this integral enables us to compute the mean square of the sums of $$d_k(n)$$ in terms of a lattice point count. This lattice point count can in turn be calculated in terms of a certain piecewise polynomial function, which we analyse. Our results suggest general conjectures for the corresponding classical problems over the integers, which agree with the few cases where the answer is known.

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

We calculate the mean and variance of sums of the M\"obius function and the indicator function of the squarefrees, in both short intervals and arithmetic progressions, in the context of the ring of polynomials over a finite field of $q$ elements, in the limit $q\to \infty$. We do this by relating the sums in question to certain matrix integrals over the unitary group, using recent equidistribution results due to N. Katz, and then by evaluating these integrals. In many cases our results mirror what is either known or conjectured for the corresponding problems involving sums over the integers, which have a long history. In some cases there are subtle and surprising differences. The ranges over which our results hold is significantly greater than those established for the corresponding problems in the number field setting., Comment: Added references and corrected misprints

We establish a connection between the conjectural two-over-two ratios formula for the Riemann zeta-function and a conjecture concerning correlations of a certain arithmetic function. Specifically, we prove that the ratios conjecture and the arithmetic correlations conjecture imply the same result. This casts a new light on the underpinnings of the ratios conjecture, which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe.

We investigate the approximation of quadratic Dirichlet $L$-functions over function fields by truncations of their Euler products. We first establish representations for such $L$-functions as products over prime polynomials times products over their zeros. This is the hybrid formula in function fields. We then prove that partial Euler products are good approximations of an $L$-function away from its zeros, and that, when the length of the product tends to infinity, we recover the original $L$-function. We also obtain explicit expressions for the arguments of quadratic Dirichlet $L$-functions over function fields and for the arguments of their partial Euler products. In the second part of the paper we construct, for each quadratic Dirichlet $L$-function over a function field, an auxiliary function based on the approximate functional equation that equals the $L$-function on the critical line. We also construct a parametrized family of approximations of these auxiliary functions, prove the Riemann hypothesis holds for them, and that their zeros are related to those of the associated $L$-function. Finally, we estimate the counting function for the zeros of this family of approximations, show that these zeros cluster near those of the associated $L$-function, and that, when the parameter is not too large, almost all the zeros of the approximations are simple., 23 pages

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

In this note we give a combinatorial and non-computational proof of the asymptotics of the integer moments of the moments of the characteristic polynomials of Haar distributed unitary matrices as the size of the matrix goes to infinity. This is achieved by relating these quantities to a lattice point count problem. Our main result is a new explicit expression for the leading order coefficient in the asymptotic as a volume of a certain region involving continuous Gelfand-Tsetlin patterns with constraints., Comment: Minor improvements throughout. To appear RMTA

We prove that translationally invariant Hamiltonians of a chain of $n$ qubits with nearest-neighbour interactions have two seemingly contradictory features. Firstly in the limit $n\rightarrow\infty$ we show that any translationally invariant Hamiltonian of a chain of $n$ qubits has an eigenbasis such that almost all eigenstates have maximal entanglement between fixed-size sub-blocks of qubits and the rest of the system; in this sense these eigenstates are like those of completely general Hamiltonians (i.e. Hamiltonians with interactions of all orders between arbitrary groups of qubits). Secondly in the limit $n\rightarrow\infty$ we show that any nearest-neighbour Hamiltonian of a chain of $n$ qubits has a Gaussian density of states; thus as far as the eigenvalues are concerned the system is like a non-interacting one. The comparison applies to chains of qubits with translationally invariant nearest-neighbour interactions, but we show that it is extendible to much more general systems (both in terms of the local dimension and the geometry of interaction). Numerical evidence is also presented which suggests that the translational invariance condition may be dropped in the case of nearest-neighbour chains., Comment: Updated figures, as accepted in 'Communications in Mathematical Physics' on 5 January 2015

The aim of this chapter is to motivate and describe some recent developments concerning the applications of random matrix theory to problems in number theory. The first section provides a brief and rather selective introduction to the theory of the Riemann zeta function, in particular to those parts needed to understand the connections with random matrix theory. The second section focuses on the value distribution of the zeta function on its critical line, specifically on recent progress in understanding the extreme value statistics gained through a conjectural link to log–correlated Gaussian random fields and the statistical mechanics of glasses. The third section outlines some number-theoretic problems that can be resolved in function fields using random matrix methods. In this latter case, random matrix theory provides the only route we currently have for calculating certain important arithmetic statistics rigorously and unconditionally.

We extend to the function field setting the heuristic previously developed, by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments and ratios of $L$-functions defined over number fields. Specifically, we give a heuristic for the moments and ratios of a family of $L$-functions associated with hyperelliptic curves of genus $g$ over a fixed finite field $\mathbb{F}_{q}$ in the limit as $g\rightarrow\infty$. Like in the number field case, there is a striking resemblance to the corresponding formulae for the characteristic polynomials of random matrices. As an application, we calculate the one-level density for the zeros of these $L$-functions., Comment: 40 pages

The first and second moments are established for the family of quadratic Dirichlet L–functions over the rational function field at the central point s = 1 2 where the character χ is defined by the Legendre symbol for polynomials over finite fields and runs over all monic irreducible polynomials P of a given odd degree. Asymptotic formulae are derived for fixed finite fields when the degree of P is large. The first moment obtained here is the function field analogue of a result due to Jutila in the number–field setting. The approach is based on classical analytical methods and relies on the use of the analogue of the approximate functional equation for these L–functions.

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.

In this series we examine the calculation of the 2kth moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper begins 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 length up to $$T^3$$ T 3 with divisor functions as coefficients.

We obtain an asymptotic formula for the first moment of quadratic Dirichlet L-functions over the rational function field at the central point s = 1 2 . Specifically, we compute the expected value of L ( 1 2 , χ ) for an ensemble of hyperelliptic curves of genus g over a fixed finite field as g → ∞ . Our approach relies on the use of the analogue of the approximate functional equation for such L-functions. The results presented here are the function field analogues of those obtained previously by Jutila in the number-field setting and are consistent with recent general conjectures for the moments of L-functions motivated by Random Matrix Theory.

We resolve a function field version of two conjectures concerning the variance of the number of primes in short intervals (Goldston and Montgomery) and in arithmetic progressions (Hooley). A crucial ingredient in our work are recent equidistribution results of N. Katz.

Quantum graphs are commonly used as models of complex quantum systems, for example molecules, networks of wires, and states of condensed matter. We consider quantum statistics for indistinguishable spinless particles on a graph, concentrating on the simplest case of abelian statistics for two particles. In spite of the fact that graphs are locally one-dimensional, anyon statistics emerge in a generalized form. A given graph may support a family of independent anyon phases associated with topologically inequivalent exchange processes. In addition, for sufficiently complex graphs, there appear new discrete-valued phases. Our analysis is simplified by considering combinatorial rather than metric graphs -- equivalently, a many-particle tight-binding model. The results demonstrate that graphs provide an arena in which to study new manifestations of quantum statistics. Possible applications include topological quantum computing, topological insulators, the fractional quantum Hall effect, superconductivity and molecular physics., 21 pages, 6 figures

It is believed that, in the limit as the conductor tends to infinity, correlations between the zeros of elliptic curve L-functions averaged within families follow the distribution laws of the eigenvalues of random matrices drawn from the orthogonal group. For test functions with restricted support, this is known to be the true for the one- and two-level densities of zeros within the families studied to date. However, for finite conductor Miller's experimental data reveal an interesting discrepancy from these limiting results. Here we use the L-functions ratios conjectures to calculate the 1-level density for the family of even quadratic twists of an elliptic curve L-function for large but finite conductor. This gives a formula for the leading and lower order terms up to an error term that is conjectured to be significantly smaller. The lower order terms explain many of the features of the zero statistics for relatively small conductor and model the very slow convergence to the infinite conductor limit. However, our main observation is that they do not capture the behaviour of zeros in the important region very close to the critical point and so do not explain Miller's discrepancy. This therefore implies that a more accurate model for statistics near to this point needs to be developed.

We study the resonance eigenstates of a particular quantization of the open baker map. For any admissible value of Planck's constant, the corresponding quantum map is a subunitary matrix, and the nonzero component of its spectrum is contained inside an annulus in the complex plane, $|z_{min}|\leq |z|\leq |z_{max}|$. We consider semiclassical sequences of eigenstates, such that the moduli of their eigenvalues converge to a fixed radius $r$. We prove that, if the moduli converge to $r=|z_{max}|$, then the sequence of eigenstates converges to a fixed phase space measure $��_{max}$. The same holds for sequences with eigenvalue moduli converging to $|z_{min}|$, with a different limit measure $��_{min}$. Both these limiting measures are supported on fractal sets, which are trapped sets of the classical dynamics. For a general radius $|z_{min}|< r < |z_{max}|$, we identify families of eigenstates with precise self-similar properties., 32 pages, 2 figures

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., 37 pages, 4 figures

We compute the moments of the characteristic polynomials of random orthogonal and symplectic matrices, defined by averages with respect to Haar measure on SO(2N) and USp(2N), to leading order as N → ∞, on the unit circle as functions of the angle θ measured from one of the two symmetry points in the eigenvalue spectrum $${\{\exp(\pm i \theta_n)\}_{1\le n \le N}}$$ . Our results extend previous formulae that relate just to the symmetry points, i.e. to θ = 0. Local spectral statistics are expected to converge to those of random unitary matrices in the limit as N → ∞ when θ is fixed, and to show a transition from the orthogonal or symplectic to the unitary forms on the scale of the mean eigenvalue spacing: if θ = π y/N they become functions of y in the limit when N → ∞. We verify that this is true for the spectral two-point correlation function, but show that it is not true for the moments of the characteristic polynomials, for which the leading order asymptotic approximation is a function of θ rather than y. Symmetry points therefore influence the moments asymptotically far away on the scale of the mean eigenvalue spacing. We also investigate the moments of the logarithms of the characteristic polynomials in the same context. The moments of the characteristic polynomials of random matrices are conjectured to be related to the moments of families of L-functions. Previously, moments at the symmetry point θ = 0 have been related to the moments of families of L-functions evaluated at the centre of the critical strip. Our results motivate general conjectures for the moments of orthogonal and symplectic families of L-functions evaluated at a fixed height t up the critical line. These conjectures suggest that the symmetry of the non-trivial zeros of the L-functions influences the moments asymptotically far, on the scale of the mean zero spacing, from the centre of the critical strip. We verify that the second moments of real quadratic Dirichlet L-functions and a family of automorphic L-functions are consistent with our conjectures.

Hybrid Euler-Hadamard products have previously been studied for the Riemann zeta function on its critical line and for Dirichlet L-functions in the context of the calculation of moments and connections with Random Matrix Theory. According to the Katz-Sarnak classification, these are believed to represent families of L-function with unitary symmetry. We here extend the formalism to families with orthogonal & symplectic symmetry. Specifically, we establish formulae for real quadratic Dirichlet L-functions and for the L-functions associated with primitive Hecke eigenforms of weight 2 in terms of partial Euler and Hadamard products. We then prove asymptotic formulae for some moments of these partial products and make general conjectures based on results for the moments of characteristic polynomials of random matrices.

We address the question as to why, in the semiclassical limit, classically chaotic systems generically exhibit universal quantum spectral statistics coincident with those of Random Matrix Theory. To do so, we use a semiclassical resummation formalism that explicitly preserves the unitarity of the quantum time evolution by incorporating duality relations between short and long classical orbits. This allows us to obtain both the non-oscillatory and the oscillatory contributions to spectral correlation functions within a unified framework, thus overcoming a significant problem in previous approaches. In addition, our results extend beyond the universal regime to describe the system-specific approach to the semiclassical limit., Comment: 10 pages, no figures

We study the 2kth power moment of Dirichlet L-functions L(s, χ) at the centre of the critical strip (s = 2 ), where the average is over all primitive characters χ (mod q). We extend to this case the hybrid Euler–Hadamard product results of Gonek, Hughes and Keating for the Riemann zeta function. This allows us to recover conjectures for the moments based on random matrix models, incorporating the arithmetical terms in a natural way.

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 present an algorithm, based on the explicit formula for $L$-functions and conditional on GRH, for proving that a given integer is squarefree with little or no knowledge of its factorization. We analyze the algorithm both theoretically and practically, and use it to prove that several RSA challenge numbers are not squarefull., 31 pages, 3 figures, latest version

We compute critical properties of a general class of quantum spin chains which are quadratic in the Fermi operators and can be solved exactly under certain symmetry constraints related to the classical compact groups $U(N)$, $O(N)$ and $Sp(2N)$. In particular we calculate critical exponents $s$, $\nu$ and $z$, corresponding to the energy gap, correlation length and dynamic exponent respectively. We also compute the ground state correlators $\left\langle \sigma^{x}_{i} \sigma^{x}_{i+n} \right\rangle_{g}$, $\left\langle \sigma^{y}_{i} \sigma^{y}_{i+n} \right\rangle_{g}$ and $\left\langle \prod^{n}_{i=1} \sigma^{z}_{i} \right\rangle_{g}$, all of which display quasi-long-range order with a critical exponent dependent upon system parameters. Our approach establishes universality of the exponents for the class of systems in question., Comment: 14 pages

We establish the equivalence of conjectures concerning the pair correlation of zeros of L-functions in the Selberg class and the variances of sums of a related class of arithmetic functions over primes in short intervals. This extends the results of Goldston and Montgomery [‘Pair correlation of zeros and primes in short intervals’, Analytic number theory and Diophantine problems (Stillwater, 1984), Progress in Mathematics 70 (1987) 183–203] and Montgomery and Soundararajan [‘Primes in short intervals’, Comm. Math. Phys. 252 (2004) 589–617] for the Riemann zeta-function to other L-functions in the Selberg class. Our approach is based on the statistics of the zeros because the analogue of the Hardy–Littlewood conjecture for the auto-correlation of the arithmetic functions we consider is not available in general. One of our main findings is that the variances of sums of these arithmetic functions over primes in short intervals have a different form when the degree of the associated L-functions is 2 or higher to that which holds when the degree is 1 (for example, the Riemann zeta-function). Specifically, when the degree is 2 or higher, there are two regimes in which the variances take qualitatively different forms, whilst in the degree-1 case there is a single regime.

Conjectured links between the distribution of values taken by the characteristic polynomials of random orthogonal matrices and that for certain families of L-functions at the center of the critical strip are used to motivate a series of conjectures concerning the value distribution of the Fourier coefficients of halfintegral- weight modular forms related to these L-functions. Our conjectures may be viewed as being analogous to the Sato–Tate conjecture for integral-weight modular forms. Numerical evidence is presented in support of them.

We give a new heuristic for all of the main terms in the integral moments of various families of primitive $L$-functions. 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 $L$-functions have a spectral interpretation, and that their value distributions can be modelled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.

We have computed quantum wavefunctions in the high-energy (semiclassical) regime in a system—the stadium billiard with leads allowing particles to enter and escape—exemplifying chaotic scattering. The results exhibit a structure associated with classical paths that is dramatically more pronounced than the scars due to periodic orbits seen in bound systems. Moreover, this structure is seen at all energies. Our results differ significantly from those previously conjectured on the basis of computations in the low-energy regime. The dominant role played by short classical paths is explained by a semiclassical theory based on the analytic structure of the Green function. This is verified by a direct semiclassical computation of wavefunctions using classical scattering trajectories.

Recent results concerning the semiclassical influence of bifurcating classical periodic orbits on quantum fluctuation statistics are reviewed. These point to the appearance of a new class of universal scaling exponents that characterize the divergence of spectral and wavefunction statistics in systems where chaotic and regular dynamics coexist (i.e. in systems with a mixed phase space) in the limit asћ→ 0.

In recent years there has been a growing interest in connections between the statistical properties of number theoretical L-functions and random matrix theory. We review the history of these connections, some of the major achievements and a number of applications.

We review observations of bubble size distributions (BSDs) generated during explosive volcanic eruptions and laboratory explosions, as inferred from vesicle size distributions found in the end products. Unimodal, polymodal, exponential and power law BSDs are common, even in the absence of coalescence, and both power law and exponential distributions have been generated in the same eruption. To date theoretical models have proposed incompatible mechanisms for producing the various distributions. We here present a unifying mechanism. Data from our laboratory analogue experiments suggest that power law distributions are associated with highly non-equilibrium degassing. A numerical model is developed in which bubbles nucleate repeatedly and grow in the spaces between those of previous generations, where, in a non-equilibrium degassing scenario, the volatile concentration remains high. This process causes the BSD to evolve from unimodal, through exponential, into a power law. The exponent of the power law is a measure of the number of nucleation events, or the duration of the nucleation period compared with the timescale of bubble growth. The mathematical inevitability of the evolution from unimodal (Poissonian) to power law is discussed. The findings may resolve the apparent contradiction between the equilibrium degassing conduit flow models and the non-equilibrium degassing conditions derived from bubble growth models of explosive volcanic eruptions. The process of ongoing nucleation is the mechanism whereby the volcanic system maintains near-equilibrium in the case of rapid depressurisation and slow volatile diffusion.

It was recently shown (Keating & Prado, {\it Proc. R. Soc. Lond. A} {\bf 457}, 1855-1872 (2001)) that, in the semiclassical limit, the scarring of quantum eigenfunctions by classical periodic orbits in chaotic systems may be dramatically enhanced when the orbits in question undergo bifurcation. Specifically, a bifurcating orbit gives rise to a scar with an amplitude that scales as $\hbar^{\alpha}$ and a width that scales as $\hbar^{\omega}$, where $\alpha$ and $\omega$ are bifurcation-dependent scar exponents whose values are typically smaller than those ($\alpha=\omega=1/2$) associated with isolated and unstable periodic orbits. We here analyze the influence of bifurcations on the autocorrelation function of quantum eigenstates, averaged with respect to energy. It is shown that the length-scale of the correlations around a bifurcating orbit scales semiclassically as $\hbar^{1-\alpha}$, where $\alpha$ is the corresponding scar amplitude exponent. This imprint of bifurcations on quantum autocorrelations is illustrated by numerical computations for a family of perturbed cat maps., Comment: 25 pages, 11 postscript figures. Some of the pictures are included in low resolution only. For a version with pictures in high resolution see http://www.physik.uni-ulm.de/theo/qc/

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.

The negative moments of spectral determinants ∏n|1-(E + iδ)/En|-k diverge when δ→0 as δ-ν(k). For a spectrum with equally distributed levels, the exponent ν(k) = k-1. For random-matrix ensembles, with parameters β = 1 (orthogonal), 2 (unitary), 4 (symplectic), we argue that the divergences for each k are determined by competitions between near-degenerate level clusters whose sizes depend on k, and we conjecture that ν(k) = int[(k-1)/β + 1]((k-1 + ½β)-½βint[(k-1)/β + 1]). For Poisson-distributed levels, unrestricted clustering leads to the δ-divergence of the moments increasing with the number N of levels in the interval considered, and ν(k) = N(k-1).

We calculate the force operator for a charged particle in the field of an Aharonov-Bohm flux line. Formally this is the Lorentz force, with the magnetic field operator modified to include quantum corrections due to anomalous commutation relations. For stationary states, the magnitude of the force is proportional to the product of the wavenumber k with the amplitudes of the `pinioned' components, the two angular momentum components whose azimuthal quantum numbers are closest to the flux parameter α. The direction of the force depends on the relative phase of the pinioned components. For paraxial beams, the transverse component of our expression gives an exact version of Shelankov's formula (Shelankov A 1998 Europhys. Lett. 43 623-8), while the longitudinal component gives the force along the beam. Nonstationary states are treated by integrating the force operator in time to obtain the impulse operator. Expectation values of the impulse are calculated for two kinds of wavepacket. For slow wavepackets, which spread faster than they move, the impulse is inversely proportional to the distance from the flux line. For fast wavepackets, which spread only negligibly before their closest approach to the flux line, the impulse is proportional to the probability density transverse to the incident direction evaluated at the flux line. In this case, the transverse component of the impulse gives a wavepacket analogue of Shelankov's formula. The direction of the impulse for both kinds of wavepacket is flux dependent. We give two derivations of the force and impulse operators, the first a simple derivation based on formal arguments, and the second a rigorous calculation of wavepacket expectation values. We also show that the same expressions for the force and impulse are obtained if the flux line is enclosed in an impenetrable cylinder, or distributed uniformly over a flux cylinder, in the limit that the radius of the cylinder goes to zero.