Correlations between zeros of a random polynomial

We obtain exact analytical expressions for correlations between real zeros of the Kac random polynomial. We show that the zeros in the interval (−1, 1) are asymptotically independent of the zeros outside of this interval, and that the straightened zeros have the same limit-translation-invariant correlations. Then we calculate the correlations between the straightened zeros of theO(1) random polynomial.     

Emergence of Heavy-Tailed Distributions in a Random Multiplicative Model Driven by a Gaussian Stochastic Process

We consider a random multiplicative stochastic process with multipliers given by the exponential of a Brownian motion. The positive integer moments of the distribution function can be computed exactly, and can be represented as the grand partition function of an equivalent lattice gas with attractive 2-body interactions. The numerical results for the positive integer moments display a sharp transition at a critical value of the model parameters, which corresponds to a phase transition in the equivalent lattice gas model. The shape of the terminal distribution changes suddenly at the critical point to a heavy-tailed distribution. The transition can be related to the position of the complex zeros of the grand partition function of the lattice gas, in analogy with the Lee, Yang picture of phase transitions in statistical mechanics. We study the properties of the equivalent lattice gas in the thermodynamical limit, which corresponds to the continuous time limit of the random multiplicative model, and derive the asymptotics of the approach to the continuous time limit. The results can be generalized to a wider class of random multiplicative processes, driven by the exponential of a Gaussian stochastic process.     

Density of Complex Zeros of a System of Real Random Polynomials

We study the density of complex zeros of a system of real random SO(m+1) polynomials in m variables. We show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of complex zeros in the complex coefficients case. We also show that the behavior the scaled density of complex zeros near ℝ ^m of the system of real random polynomials is different in the m≥2 case than in the m=1 case: the density approaches infinity instead of tending linearly to zero.     

SU(1, 1) Random Polynomials

We study statistical properties of zeros of random polynomials and random analytic functions associated with the pseudoeuclidean group of symmetries SU(1, 1), by utilizing both analytical and numerical techniques. We first show that zeros of the SU(1, 1) random polynomial of degree N are concentrated in a narrow annulus of the order of N ^–1 around the unit circle on the complex plane, and we find an explicit formula for the scaled density of the zeros distribution along the radius in the limit N. Our results are supported through various numerical simulations. We then extend results of Hannay⁽¹⁾ and Bleher et al. ⁽²⁾ to derive different formulae for correlations between zeros of the SU(1, 1) random analytic functions, by applying the generalized Kac–Rice formula. We express the correlation functions in terms of some Gaussian integrals, which can be evaluated combinatorially as a finite sum over Feynman diagrams or as a supersymmetric integral. Due to the SU(1, 1) symmetry, the correlation functions depend only on the hyperbolic distances between the points on the unit disk, and we obtain an explicit formula for the two point correlation function. It displays quadratic repulsion at small distances and fast decay of correlations at infinity. In an appendix to the paper we evaluate correlations between the outer zeros |z _j |>1 of the SU(1, 1) random polynomial, and we prove that the inner and outer zeros are independent in the limit when the degree of the polynomial goes to infinity.     

Theory and calculations for a spin glass

We obtain the properties of a mean-field spin-glass (in which the bonds connecting each spin to every other spin are “frozen-in” with random signs), by locating the zeros of the partition function in the complex T plane. For N = 5 and 9 spins, we obtain the relevant polynomials and zeros explicitly, and the resulting thermodynamic properties (free energy, specific heat, magnetic susceptibility, etc.). We then analyze the properties of such a system in the thermodynamic limit N → ∞, where it is impossible to obtain the polynomials directly but where the presumed location of the zeros can be usefully construed. In this limit, the thermodynamic functions are obtainable as functions of the distribution functions of monopoles, quadrupoles, and possibly higher-order poles.     

Density of Complex Critical Points of a Real Random <Emphasis Type="Italic">SO</Emphasis>(<Emphasis Type="Italic">m</Emphasis>+1) Polynomial

We study the density of complex critical points of a real random SO(m+1) polynomial in m variables. In a previous paper (Macdonald in J. Stat. Phys. 136(5):807, 2009), the author used the Poincaré-Lelong formula to show that the density of complex zeros of a system of these real random polynomials rapidly approaches the density of complex zeros of a system of the corresponding complex random polynomials, the SU(m+1) polynomials. In this paper, we use the Kac-Rice formula to prove an analogous result: the density of complex critical points of one of these real random polynomials rapidly approaches the density of complex critical points of the corresponding complex random polynomial. In one variable, we give an exact formula and a scaling limit formula for the density of critical points of the real random SO(2) polynomial as well as for the density of critical points of the corresponding complex random SU(2) polynomial.     

Yang–Lee zeros of the Ising model on random graphs of non planar topology

We obtain in a closed form the 1/N² contribution to the free energy of the two Hermitian N×N random matrix model with nonsymmetric quartic potential. From this result, we calculate numerically the Yang–Lee zeros of the 2D Ising model on dynamical random graphs with the topology of a torus up to n=16 vertices. They are found to be located on the unit circle on the complex fugacity plane. In order to include contributions of even higher topologies we calculated analytically the nonperturbative (sum over all genus) partition function of the model for the special cases of N=1,2 and graphs with n≤20 vertices. Once again the Yang–Lee zeros are shown numerically to lie on the unit circle on the complex fugacity plane. Our results thus generalize previous numerical results on random graphs by going beyond the planar approximation and strongly indicate that there might be a generalization of the Lee–Yang circle theorem for dynamical random graphs.     

Exact results for deterministic cellular automata traffic models

We present a rigorous derivation of the flow at arbitrary time in a deterministic cellular automaton model of traffic flow. The derivation employs regularities in preimages of blocks of zeros, reducing the problem of preimage enumeration to a well-known lattice path counting problem. Assuming infinite lattice size and random initial configuration, the flow can be expressed in terms of generalized hypergeometric function. We show that the steady-state limit agrees with previously published results.     

Nodal lines,ergodicity and complex numbers

This article reviews two rigorous results about the complex zeros of eigenfunctions of the Laplacian, that is, the zeros of the analytic continuation of the eigenfunctions to the complexification of the underlying space. Such a complexification of the problem is analogous to studying the complex zeros of polynomials with real coefficients. The first result determines the limit distribution of complex zeros of `ergodic eigenfunctions' such as eigenfunctions of classically chaotic systems. The second result determines the expected distribution of complex zeros for complexifications of Gaussian random waves adapted to the Riemannian manifold. The resulting distribution is the same in both cases. It is singular along the set of real points.     

Distribution of zeros in Ising and gauge models

We discuss some features of Ising and gauge systems in the complex temperature plane. The distribution of zeros of the partition function enables one to study critical properties in a way complementary to the methods using real values. Data on small lattices confirm this picture. Nearby complex singularities seem to exhibit a universal behavior which might have some relation with a model of random surfaces.     

Lee–Yang zeros in the Rydberg atoms

Lee–Yang (LY) zeros play a fundamental role in the formulation of statistical physics in terms of (grand) partition functions, and assume theoretical significance for the phenomenon of phase transitions. In this paper, motivated by recent progress in cold Rydberg atom experiments, we explore the LY zeros in classical Rydberg blockade models. We find that the distribution of zeros of partition functions for these models in one dimension (1d) can be obtained analytically. We prove that all the LY zeros are real and negative for such models with arbitrary blockade radii. Therefore, no phase transitions happen in 1d classical Rydberg chains. We investigate how the zeros redistribute as one interpolates between different blockade radii. We also discuss possible experimental measurements of these zeros.     

In Support of n-Correlation

In this paper we examine n-correlation for either the eigenvalues of a unitary group of random matrices or for the zeros of a unitary family of L-functions in the important situation when the correlations are detected via test functions whose Fourier transforms have limited support. This problem first came to light in the work of Rudnick and Sarnak in their study of the n-correlation of zeros of a fairly general automorphic L-function. They solved the simplest instance of this problem when the total support was most severely limited, but had to work extremely hard to show their result matched random matrix theory in the appropriate limit. This is because they were comparing their result to the familiar determinantal expressions for n-correlation that arise naturally in random matrix theory. In this paper we deal with arbitrary support and show that there is another expression for the n-correlation of eigenvalues that translates easily into the number theory case and allows for immediate identification of which terms survive the restrictions placed on the support of the test function.     

Radiation zeros in high-energy e^+e^- annihilation into hadrons

The process contains radiation zeros, i.e. configurations of the four–momenta for which the scattering amplitude vanishes. We calculate the positions of these zeros for –quark and –quark production and assess the feasibility of identifying the zeros in experiments at high energies. The radiation zeros are shown to occur also for massive quarks, and we discuss how the final state may offer a particularly clean environment in which to observe them. Received: 16 December 1997 / Published online: 10 March 1998     

Nodal densities of planar gaussian random waves

The nodal densities of gaussian random functions, modelling various physical systems including chaotic quantum eigenfunctions and optical speckle patterns, are reviewed. The nodal domains of isotropically random real and complex functions are formulated in terms of their Minkowski functionals, and their correlations and spectra are discussed. The results on the statistical densities of the zeros of the real and complex functions, and their derivatives, in two dimensions are reviewed. New results are derived on the nodal domains of the hessian determinant (gaussian curvature) of two-dimensional random surfaces.     

General theory of lee-yang zeros in models with first-order phase transitions

We present a general, rigorous theory of Lee-Yang zeros for models with first-order phase transitions that admit convergent contour expansions. We derive formulas for the positions and the density of the zeros. In particular, we show that, for models without symmetry, the curves on which the zeros lie are generically not circles, and can have topologically nontrivial features, such as bifurcation. Our results are illustrated in three models in a complex field: the low-temperature Ising and Blume-Capel models, and the q-state Potts model for large q.     

Distribution of zeros of the partition function of the lattice gases of fisher and of temperley

The distributions of zeros of the partition function of lattice gas models of Fisher (at some temperature) and of Temperley are obtained. The former is a closed loop crossing the real axis and the latter is a part of the negative real axis.     

Do amplitude zeros persist in higher order?

We consider the phenomenon of amplitude zeros, first discovered in the process d u→W^?γ. Using spin-0 particles, we find that the zeros persist for radiation from internal bubbled. However, when we consider the 1-loop correction to the scalar three-point function (with a photon attached in all possible ways), it is shown, by an explicit calculation, that amplitude zeros do not persist in general.     

Large Deviations for Zeros of <Emphasis Type="Italic">P</Emphasis>(<Emphasis Type="Italic">ϕ</Emphasis>)<Subscript>2</Subscript> Random Polynomials

We extend the results of (Zeitouni and Zelditch in Int. Math. Res. Not. 2010(20):3939–3992, 2010) on LDPs (large deviations principles) for the empirical measures

$Z_s: = \frac{1}{N} \sum_{\zeta: s(\zeta) = 0} \delta_{\zeta}, \quad (N: = \# \{\zeta: s(\zeta) = 0\})$

of zeros of Gaussian random polynomials s in one variable to P(?)₂ random polynomials. The speed and rate function are the same as in the associated Gaussian case. It follows that the expected distribution of zeros in the P(?)₂ ensembles tends to the same equilibrium measure as in the Gaussian case.