arXiv subjects

Theodoros Assiotis

Publications and source records attributed to Theodoros Assiotis.

At least 19 recordsLinked to original sources

arXiv2026

Uniqueness for DLR equations of $\mathsf{Sine}_\beta$

We prove that, for all $\beta>0$, the Dobrushin-Lanford-Ruelle (DLR) equations for the unit-intensity $\mathsf{Sine}_\beta$ point process have a unique stationary solution under a particular finite electric energy condition. Uniqueness fails if this condition is dropped. This answers a question of Dereudre-Hardy-Lebl\'e-Ma\"ida and gives a canonical statistical physics characterisation of $\mathsf{Sine}_\beta$. The main technical ingredient is a relative entropy estimate which allows us to compare the conditional distribution in a finite box of a DLR solution to a corresponding circular $\beta$ ensemble.

math.PR↗

arXiv2026

On non-monotonicity of logarithmic energy for random matrices

We construct a finite-energy Wigner counterexample and a concrete one-parameter family of Gaussian-regularised Bernoulli entry laws yielding counterexamples to the conjectural dimensional monotonicity of the quadratically penalised logarithmic energy for mean empirical spectral distributions by Chafa\"i, Dadoun, and Youssef.

math.PR↗

arXiv2026

On the heat flow conjecture for random matrices

We prove general cases of the heat flow conjecture for random matrices of Hall-Ho. In particular, the empirical measure of the zeros of the heat-flow-evolved characteristic polynomial of a complex Ginibre matrix converges almost surely to the semicircle law.

math.PR↗

arXiv2026

Collision and non-collision for diffusions on configuration space

We develop criteria for collision and non-collision of reversible infinitely many interacting diffusion processes in the real line. The approach is potential-theoretic and is based on capacity estimates for symmetric Dirichlet forms on the configuration space. Our main results are model-independent in the sense that no determinantal or Pfaffian structure, prescribed interaction potential, or explicit labelled stochastic differential equation is required. The non-collision criterion involves only the second and third correlation functions of the reversible measure, whereas the collision criterion is based on a local lower bound for a finite-volume conditional density of the reversible measure. As an application, we identify the sharp collision threshold for the diffusion associated with the $\mathsf{Sine}_\beta$-symmetric Dirichlet form: the collision set is polar if and only if $\beta\ge 1$. This provides an infinite-particle counterpart of the classical collision threshold for the finite-particle Dyson Brownian motion with inverse temperature $\beta$.

math.PR↗

arXiv2026

Non-colliding space-time inhomogeneous Markov chains

We establish the explicit leading order asymptotics, with a quantitative error bound, of tail probabilities of collision times for a class of integrable space-time inhomogeneous Markov chains, in discrete and continuous time. The corresponding process conditioned not to intersect arises in interacting particle systems with local push-block interactions thereby confirming a recent prediction. The generic discrete nature of the spatial inhomogeneities rules out powerful coupling-with-Brownian-motion techniques, so our proof strategy proceeds instead via a novel steepest-descent analysis combined with a Karlin--McGregor semigroup expansion in terms of dominant-index contributions.

math.PR↗

arXiv2026

$\mathsf{GL}_N(\mathbb{C})$ Brownian motion and stochastic PDE on entire functions

We construct the full edge scaling limit of the singular values of Brownian motion on the general linear group $\mathsf{GL}_N(\mathbb{C})$ starting from general conditions. We show that the limiting paths solve an infinite system of SDE with log-interaction and have a Gibbs resampling property with exponential Brownian bridges. Moreover, we show that the evolution of the limiting rescaled reverse characteristic polynomial solves a stochastic partial differential equation with a non-linear multiplicative noise and linear drift. From a special initial condition the resulting line ensemble coincides, in logarithmic coordinates, with a line ensemble constructed by Ahn which arises as a universal scaling limit of singular values of products of random matrices. We prove some analogous results on the evolution of limiting characteristic polynomials for two models whose stationary measures are given by the Hua-Pickrell and Bessel stochastic zeta functions respectively.

math.PR↗

arXiv2026

Moments of C$\beta$E field partition function, $\mathsf{Sine}_{\beta}$ correlations and stochastic zeta

We prove a conjecture of Fyodorov and Keating on the supercritical moments of the partition function of the C$\beta$E field or equivalently the supercritical moments of moments of the characteristic polynomial of the C$\beta$E ensemble for general $\beta>0$ and general real moment exponents. Moreover, we give the first expression for all correlation functions of the $\mathsf{Sine}_\beta$ point process for all $\beta>0$. The main object behind both results is the Hua-Pickrell stochastic zeta function introduced by Li and Valk\'{o}.

math.PR↗

arXiv2025

Joint Moments of Characteristic Polynomials from the Orthogonal and Unitary Symplectic Groups

We establish asymptotic formulae for general joint moments of characteristic polynomials and their higher-order derivatives associated with matrices drawn randomly from the groups $\mathrm{USp}(2N)$ and $\mathrm{SO}(2N)$ in the limit as $N\to\infty$. This relates the leading-order asymptotic contribution in each case to averages over the Laguerre ensemble of random matrices. We uncover an exact connection between these joint moments and a solution of the $\sigma$-Painlev\'{e} V equation, valid for finite matrix size, as well as a connection between the leading-order asymptotic term and a solution of the $\sigma$-Painlev\'{e} III$'$ equation in the limit as $N \rightarrow \infty$. These connections enable us to derive exact formulae for joint moments for finite matrix size and for the joint moments of certain random variables arising from the Bessel point process in a recursive way. As an application, we provide a positive answer to a question proposed by Altu\u{g} et al.

math-ph↗

arXiv2024

ISDE with logarithmic interaction and characteristic polynomials

We consider certain random matrix eigenvalue dynamics, akin to Dyson Brownian motion, introduced by Rider and Valko. We show that from every initial condition, including ones involving coinciding coordinates, the dynamics, enhanced with more information, converge on path-space to a new infinite-dimensional Feller-continuous diffusion process. We show that the limiting diffusion solves an infinite-dimensional system of stochastic differential equations (ISDE) with logarithmic interaction. Moreover, we show convergence in the long-time limit of the infinite-dimensional dynamics starting from any initial condition to the equilibrium measure, given by the inverse points of the Bessel determinantal point process. As far as we can tell, this is: (a) the first path-space convergence result of random matrix dynamics starting from every initial condition to an infinite-dimensional Feller diffusion, (b) the first construction of solutions to an ISDE with logarithmic interaction from every initial condition for which the singular drift term can be defined at time $0$, (c) the first convergence to equilibrium result from every initial condition for an ISDE of this kind. The argument splits into two parts. The first part builds on the method of intertwiners introduced and developed by Borodin and Olshanski. The main new ingredients are a uniform, in a certain sense, approximation theorem of the spectrum of a family of random matrices indexed by an infinite-dimensional space and an extension of the method of intertwiners to deal with convergence to equilibrium. The second part introduces a new approach towards convergence of the singular drift term in the dynamics and for showing non-intersection of the limiting paths via certain ``characteristic polynomials" associated to the process. We believe variations of it will be applicable to other infinite-dimensional dynamics coming from random matrices.

math.PR↗

arXiv2024

Exchangeable arrays and integrable systems for characteristic polynomials of random matrices

The joint moments of the derivatives of the characteristic polynomial of a random unitary matrix, and also a variant of the characteristic polynomial that is real on the unit circle, in the large matrix size limit, have been studied intensively in the past twenty five years, partly in relation to conjectural connections to the Riemann zeta-function and Hardy's function. We completely settle the most general version of the problem of convergence of these joint moments, after they are suitably rescaled, for an arbitrary number of derivatives and with arbitrary positive real exponents. Our approach relies on a hidden, higher-order exchangeable structure, that of an exchangeable array. Using these probabilistic techniques, we then give a combinatorial formula for the leading order coefficient in the asymptotics of the joint moments, when the power on the characteristic polynomial itself is a positive real number and the exponents of the derivatives are integers, in terms of a finite number of finite-dimensional integrals which are explicitly computable. Finally, we develop a method, based on a class of Hankel determinants shifted by partitions, that allows us to give an exact representation of all these joint moments, for finite matrix size, in terms of derivatives of Painlev\'e V transcendents, and then for the leading order coefficient in the large-matrix limit in terms of derivatives of solutions of the $\sigma$-Painlev\'e III' equation. Equivalently, we can represent all the joint moments of power sum linear statistics of a certain determinantal point process behind this problem in terms of derivatives of $\sigma$-Painlev\'e III' transcendents. This gives an efficient way to compute all these quantities explicitly. Our methods can be used to obtain analogous results for a number of other models sharing the same features.

math.PR↗

arXiv2023

On some integrable models in inhomogeneous space

The purpose of this work is to build a framework that allows for an in-depth study of various generalisations to inhomogeneous space of models of Borodin-Ferrari, Dieker-Warren, Nordenstam, Warren-Windridge of interacting particles in interlacing arrays, both in discrete and continuous time, involving both Bernoulli and geometric jumps. The models can in addition be either time-inhomogeneous or particle-inhomogeneous. We show that the correlation functions of these models are determinantal and using this we prove a short-time asymptotic for these dynamics to the discrete Bessel point process. We moreover prove a number of closely related results. We prove that the autonomous, inhomogeneous in space and time, TASEP-like and pushTASEP-like particle systems on the edges of the array have explicit transition kernels and that from any deterministic initial condition their distributions are marginals of a determinantal measure. We prove a novel duality relation between dynamics in inhomogeneous space and dynamics with inhomogeneities on the level of the array. We extend the work of Nordenstam on the shuffling algorithm for domino tilings of the Aztec diamond and its relation to push-block dynamics in interlacing arrays to general weights on the tilings and then connect this, for a special class of weights, back to our previous results. We also consider non-intersecting walks in inhomogeneous space and time with fixed starting and end points and obtain a formula for their correlation functions, involving among other ingredients, an explicit Riemann-Hilbert problem. We then prove a limit theorem for the bottom lines in this line-ensemble, under some technical conditions. The main computational tool throughout this work is a natural generalisation of a Toeplitz matrix, that we call inhomogeneous Toeplitz-like matrix $\mathsf{T}_{\mathbf{f}}$ with (a possibly matrix-valued) symbol $\mathbf{f}$.

math.PR↗

arXiv2023

Exact solution of interacting particle systems related to random matrices

We consider one-dimensional diffusions, with polynomial drift and diffusion coefficients, so that in particular the motion can be space-inhomogeneous, interacting via one-sided reflections. The prototypical example is the well-known model of Brownian motions with one-sided collisions, also known as Brownian TASEP, which is equivalent to Brownian last passage percolation. We obtain a formula for the finite dimensional distributions of these particle systems, starting from arbitrary initial condition, in terms of a Fredholm determinant of an explicit kernel. As far as we can tell, in the space-inhomogeneous setting and for general initial condition this is the first time such a result has been proven. We moreover consider the model of non-colliding diffusions, again with polynomial drift and diffusion coefficients, which includes the ones associated to all the classical ensembles of random matrices. We prove that starting from arbitrary initial condition the induced point process has determinantal correlation functions in space and time with an explicit correlation kernel. A key ingredient in our general method of exact solution for both models is the application of the backward in time diffusion flow on certain families of polynomials constructed from the initial condition.

math.PR↗

arXiv2023

On the moments of moments of random matrices and Ehrhart polynomials

There has been significant interest in studying the asymptotics of certain generalised moments, called the moments of moments, of characteristic polynomials of random Haar-distributed unitary and symplectic matrices, as the matrix size $N$ goes to infinity. These quantities depend on two parameters $k$ and $q$ and when both of them are positive integers it has been shown that these moments are in fact polynomials in the matrix size $N$. In this paper we classify the integer roots of these polynomials and moreover prove that the polynomials themselves satisfy a certain symmetry property. This confirms some predictions from the thesis of Bailey. The proof uses the Ehrhart-Macdonald reciprocity for rational convex polytopes and certain bijections between lattice points in some polytopes.

math-ph↗

arXiv2022

Random entire functions from random polynomials with real zeros

We point out a simple criterion for convergence of polynomials to a concrete entire function in the Laguerre-Pólya ($\mathcal{LP}$) class (of all functions arising as uniform limits of polynomials with only real roots). We then use this to show that any random $\mathcal{LP}$ function can be obtained as the uniform limit of rescaled characteristic polynomials of principal submatrices of an infinite unitarily invariant random Hermitian matrix. Conversely, the rescaled characteristic polynomials of principal submatrices of any infinite random unitarily invariant Hermitian matrix converge uniformly to a random $\mathcal{LP}$ function. This result also has a natural extension to $β$-ensembles. Distinguished cases include random entire functions associated to the $β$-Sine, and more generally $β$-Hua-Pickrell, $β$-Bessel and $β$-Airy point processes studied in the literature.

math.PR↗

arXiv2022

On the singular values of complex matrix Brownian motion with a matrix drift

Let $Mat_{\mathbb{C}}(K,N)$ be the space of $K\times N$ complex matrices. Let $\mathbf{B}_t$ be Brownian motion on $Mat_{\mathbb{C}}(K,N)$ starting from the zero matrix and $\mathbf{M}\in Mat_{\mathbb{C}}(K,N)$. We prove that, with $K\ge N$, the $N$ eigenvalues of $\left(\mathbf{B}_t+t\mathbf{M}\right)^*\left(\mathbf{B}_t+t\mathbf{M}\right)$ form a Markov process with an explicit transition kernel. This generalizes a classical result of Rogers and Pitman for multidimensional Brownian motion with drift which corresponds to $N=1$. We then give two more descriptions for this Markov process. First, as independent squared Bessel diffusion processes in the wide sense, introduced by Watanabe and studied by Pitman and Yor, conditioned to never intersect. Second, as the distribution of the top row of interacting squared Bessel type diffusions in some interlacting array. The last two descriptions also extend to a general class of one-dimensional diffusions.

math.PR↗

arXiv2022

Convergence and an explicit formula for the joint moments of the Circular Jacobi $β$-Ensemble characteristic polynomial

The problem of convergence of the joint moments, which depend on two parameters $s$ and $h$, of the characteristic polynomial of a random Haar-distributed unitary matrix and its derivative, as the matrix size goes to infinity, has been studied for two decades, beginning with the thesis of Hughes. Recently, Forrester considered the analogous problem for the Circular $β$-Ensemble (C$β$E) characteristic polynomial, proved convergence and obtained an explicit combinatorial formula for the limit for integer $s$ and complex $h$. In this paper we consider this problem for a generalisation of the C$β$E, the Circular Jacobi $β$-ensemble (CJ$β$E), depending on an additional complex parameter $δ$ and we prove convergence of the joint moments for general positive real exponents $s$ and $h$. We give a representation for the limit in terms of the moments of a family of real random variables of independent interest. This is done by making use of some general results on consistent probability measures on interlacing arrays. Using these techniques, we also extend Forrester's explicit formula to the case of real $s$ and $δ$ and integer $h$. Finally, we prove an analogous result for the moments of the logarithmic derivative of the characteristic polynomial of the Laguerre $β$-ensemble.

math.PR↗

arXiv2022

On the moments of the partition function of the C$β$E field

We obtain a combinatorial formula for the positive integer moments of the partition function of the $CβE_{N}$ field, or equivalently the moments of the moments of the characteristic polynomial of the $CβE_{N}$ ensemble. We then use this formula to establish the large $N$ asymptotics of these moments in the "moment-supercritical" regime. A key role is played by Jack polynomials.

math.PR↗

arXiv2021

On the joint moments of the characteristic polynomials of random unitary matrices

We establish the asymptotics of the joint moments of the characteristic polynomial of a random unitary matrix and its derivative for general real values of the exponents, proving a conjecture made by Hughes in 2001. Moreover, we give a probabilistic representation for the leading order coefficient in the asymptotic in terms of a real-valued random variable that plays an important role in the ergodic decomposition of the Hua-Pickrell measures. This enables us to establish connections between the characteristic function of this random variable and the $σ$-Painlevé III' equation.

math.PR↗