arXiv subjects

Oleg Zaboronski

Publications and source records attributed to Oleg Zaboronski.

At least 19 recordsLinked to original sources

arXiv2026

Probabilities of rare events in product kernel aggregation: An exact formula and phase diagram

We present an exact method for calculating the large deviation function describing rare fluctuations in the number of particles for product-kernel aggregation. Starting from the master equation, we derive an exact integral representation for the probability $P(M,N,t)$ of observing $N$ particles at time $t$ starting from $M$ monomers for any finite $M, N, t$. From this, we obtain an exact expression for the exponential moment $\langle p^N\rangle$ for integer $p$. Employing a replica conjecture -- numerically validated by finite-$M$ scaling -- we extend this result to real $p \geq 0$. The convex envelope of the large deviation function, obtained via a Legendre-Fenchel transform of the exponential moment, shows singular behavior. The singular structure allows us to construct the full phase diagram of product-kernel aggregation, which contains a tricritical point, separating continuous and discontinuous transitions. We also compute the asymptotic form of the LDF for small $N/M$.

cond-mat.stat-mech↗

arXiv2026

Entrance laws for coalescing and annihilating Brownian motions

Systems of instantaneously annihilating or coalescing Brownian motions on the line are considered. The extreme points of the set of entrance laws for this process are shown to be Pfaffian point processes at all times and their kernels are identified.

math.PR↗

arXiv2024

Exact calculation of the large deviation function for $k$-nary coalescence

We study probabilities of rare events in the general coalescence process, $kA\rightarrow \ell A$, where $k>\ell$. For arbitrary $k, \ell$, by rewriting these probabilities in terms of an effective action, we derive the large deviation function describing the probability of finding $N$ particles at time $t$, when starting with $M$ particles initially. Additionally, the most probable trajectory corresponding to a fixed rare event is derived.

cond-mat.stat-mech↗

arXiv2024

The Stochastic Casimir Effect

We model the one-dimensional `classical' vacuum by a system of annihilating Brownian motions on $\mathbb{R}$ with pairwise immigration. A pair of reflecting or absorbing walls placed in such a vacuum at separation $L$ experiences an attractive force which decays exponentially with $L$. This phenomenon can be regarded as a purely classical Casimir effect for a system of interacting Brownian motions.

math.PR↗

arXiv2024

$A+A \to A$, $\; \; B+A \to A$

This paper considers the decay in particle intensities for a translation invariant two species system of diffusing and reacting particles on $\mathbb{Z}^d$ for $d \geq 3$. The intensities are shown to approximately solve modified rate equations, from which their polynomial decay can be deduced. The system illustrates that the underlying diffusion and reaction rates can influence the exact polynomial decay rates, despite the system evolving in a supercritical dimension.

math.PR↗

arXiv2024

Exact calculation of the probabilities of rare events in cluster-cluster aggregation

We develop an action formalism to calculate probabilities of rare events in cluster-cluster aggregation for arbitrary collision kernels and establish a pathwise large deviation principle with total mass being the rate. As an application, the rate function for the number of surviving particles as well as the optimal evolution trajectory are calculated exactly for the constant, sum and product kernels. For the product kernel, we argue that the second derivative of the rate function has a discontinuity. The theoretical results agree with simulations tailored to the calculation of rare events.

cond-mat.stat-mech↗

arXiv2023

Averages of products of characteristic polynomials and the law of real eigenvalues for the real Ginibre ensemble

An elementary derivation of the Borodin-Sinclair-Forrester-Nagao Pfaffian point process, which characterises the law of real eigenvalues for the real Ginibre ensemble in the large matrix size limit, uses the averages of products of characteristic polynomials. This derivation reveals a number of interesting structures associated with the real Ginibre ensemble such as the hidden symplectic symmetry of the statistics of real eigenvalues and an integral representation for the $K$-point correlation function for any $K\in \mathbb{N}$ in terms of an asymptotically exact integral over the symmetric space $U(2K)/USp(2K)$.

math-ph↗

arXiv2023

Examples of interacting particle systems on $\mathbb{Z}$ as Pfaffian point processes: coalescing branching random walks and annihilating random walks with immigration

Two classes of interacting particle systems on $\mathbb{Z}$ are shown to be Pfaffian point processes at fixed times, and for all deterministic initial conditions. The first comprises coalescing and branching random walks, the second annihilating random walks with pairwise immigration. Various limiting Pfaffian point processes on $\mathbb{R}$ are found by diffusive rescaling, including the point set process for the Brownian net.

math.PR↗

arXiv2023

Asymptotic expansions for a class of Fredholm Pfaffians and interacting particle systems

Motivated by the phenomenon of duality for interacting particle systems we introduce two classes of Pfaffian kernels describing a number of Pfaffian point processes in the `bulk' and at the `edge'. Using the probabilistic method due to Mark Kac, we prove two Szegő-type asymptotic expansion theorems for the corresponding Fredholm Pfaffians. The idea of the proof is to introduce an effective random walk with transition density determined by the Pfaffian kernel, express the logarithm of the Fredholm Pfaffian through expectations with respect to the random walk, and analyse the expectations using general results on random walks. We demonstrate the utility of the theorems by calculating asymptotics for the empty interval and non-crossing probabilities for a number of examples of Pfaffian point processes: coalescing/annihilating Brownian motions, massive coalescing Brownian motions, real zeros of Gaussian power series and Kac polynomials, and real eigenvalues for the real Ginibre ensemble.

math.PR↗

arXiv2023

Sample-path large deviations for stochastic evolutions driven by the square of a Gaussian process

Recently, a number of physical models has emerged described by a random process with increments given by a quadratic form of a fast Gaussian process. We find that the rate function which describes sample-path large deviations for such a process can be computed from the large domain size asymptotic of a certain Fredholm determinant. The latter can be evaluated analytically using a theorem of Widom which generalizes the celebrated Szegő-Kac formula to the multi-dimensional case. This provides a large class of random dynamical systems with time scale separation for which an explicit sample-path large deviation functional can be found. Inspired by problems in hydrodynamics and atmosphere dynamics, we construct a simple example with a single slow degree of freedom driven by the square of a fast multi-variate Gaussian process and analyse its large deviations functional using our general results. Even though the noiseless limit of this example has a single fixed point, the corresponding large deviations effective potential has multiple fixed points. In other words, it is the addition of noise that leads to metastability. We use the explicit answers for the rate function to construct instanton trajectories connecting the metastable states.

cond-mat.stat-mech↗

arXiv2023

A Monte Carlo algorithm to measure probabilities of rare events in cluster-cluster aggregation

We develop a biased Monte Carlo algorithm to measure probabilities of rare events in cluster-cluster aggregation for arbitrary collision kernels. Given a trajectory with a fixed number of collisions, the algorithm modifies both the waiting times between collisions, as well as the sequence of collisions, using local moves. We show that the algorithm is ergodic by giving a protocol that transforms an arbitrary trajectory to a standard trajectory using valid Monte Carlo moves. The algorithm can sample rare events with probabilities of the order of $10^{-40}$ and lower. The algorithm's effectiveness in sampling low-probability events is established by showing that the numerical results for the large deviation function of constant-kernel aggregation reproduce the exact results. It is shown that the algorithm can obtain the large deviation functions for other kernels, including gelling ones, as well as the instanton trajectories for atypical times. The dependence of the autocorrelation times, both temporal and configurational, on the different parameters of the algorithm is also characterized.

cond-mat.stat-mech↗

arXiv2022

Representations of Hecke algebras and Markov dualities for interacting particle systems

Many continuous reaction-diffusion models on $\mathbb{Z}$ (annihilating or coalescing random walks, exclusion processes, voter models) admit a rich set of Markov duality functions which determine the single time distribution. A common feature of these models is that their generators are given by sums of two-site idempotent operators. In this paper, we classify all continuous time Markov processes on $\{0,1\}^{\mathbb{Z}}$ whose generators have this property, although to simplify the calculations we only consider models with equal left and right jumping rates. The classification leads to six familiar models and three exceptional models. The generators of all but the exceptional models turn out to belong to an infinite dimensional Hecke algebra, and the duality functions appear as spanning vectors for small-dimensional irreducible representations of this Hecke algebra. A second classification explores generators built from two site operators satisfying the Hecke algebra relations. The duality functions are intertwiners between configuration and co-ordinate representations of Hecke algebras, which results in a novel co-ordinate representations of the Hecke algebra. The standard Baxterisation procedure leads to new solutions of the Young-Baxter equation corresponding to particle systems which do not preserve the number of particles.

math.PR↗

arXiv2019

Determinantal structure and bulk universality of conditional overlaps in the complex Ginibre ensemble

In these proceedings we summarise how the determinantal structure for the conditional overlaps among left and right eigenvectors emerges in the complex Ginibre ensemble at finite matrix size. An emphasis is put on the underlying structure of orthogonal polynomials in the complex plane and its analogy to the determinantal structure of $k$-point complex eigenvalue correlation functions. The off-diagonal overlap is shown to follow from the diagonal overlap conditioned on $k\geq2$ complex eigenvalues. As a new result we present the local bulk scaling limit of the conditional overlaps away from the origin. It is shown to agree with the limit at the origin and is thus universal within this ensemble.

math-ph↗

arXiv2019

On the determinantal structure of conditional overlaps for the complex Ginibre ensemble

We continue the study of joint statistics of eigenvectors and eigenvalues initiated in the seminal papers of Chalker and Mehlig. The principal object of our investigation is the expectation of the matrix of overlaps between the left and the right eigenvectors for the complex $N\times N$ Ginibre ensemble, conditional on an arbitrary number $k=1,2,\ldots$ of complex eigenvalues.These objects provide the simplest generalisation of the expectations of the diagonal overlap ($k=1$) and the off-diagonal overlap ($k=2$) considered originally by Chalker and Mehlig. They also appear naturally in the problem of joint evolution of eigenvectors and eigenvalues for Brownian motions with values in complex matrices studied by the Krakow school. We find that these expectations possess a determinantal structure, where the relevant kernels can be expressed in terms of certain orthogonal polynomials in the complex plane. Moreover, the kernels admit a rather tractable expression for all $N \geq 2$. This result enables a fairly straightforward calculation of the conditional expectation of the overlap matrix in the local bulk and edge scaling limits as well as the proof of the exact algebraic decay and asymptotic factorisation of these expectations in the bulk.

math-ph↗

arXiv2019

Sharp asymptotics for Fredholm Pfaffians related to interacting particle systems and random matrices

It has been known since the pioneering paper of Mark Kac, that the asymptotics of Fredholm determinants can be studied using probabilistic methods. We demonstrate the efficacy of Kac' approach by studying the Fredholm Pfaffian describing the statistics of both non-Hermitian random matrices and annihilating Brownian motions. Namely, we establish the following two results. Firstly, let $\sqrt{N}+λ_{max}$ be the largest real eigenvalue of a random $N\times N$ matrix with independent $N(0,1)$ entries (the `real Ginibre matrix'). Consider the limiting $N\rightarrow \infty$ distribution $\mathbb{P}[λ_{max}<-L]$ of the shifted maximal real eigenvalue $λ_{max}$. Then \[ \lim_{L\rightarrow \infty} e^{\frac{1}{2\sqrt{2π}}ζ\left(\frac{3}{2}\right)L} \mathbb{P}\left(λ_{max}<-L\right) =e^{C_e}, \] where $ζ$ is the Riemann zeta-function and \[ C_e=\frac{1}{2}\log 2+\frac{1}{4π}\sum_{n=1}^{\infty}\frac{1}{n} \left(-π+\sum_{m=1}^{n-1}\frac{1}{\sqrt{m(n-m)}}\right). \] Secondly, let $X_t^{(max)}$ be the position of the rightmost particle at time $t$ for a system of annihilating Brownian motions (ABM's) started from every point of $\mathbb{R}_{-}$. Then \[ \lim_{L\rightarrow \infty} e^{\frac{1}{2\sqrt{2π}}ζ\left(\frac{3}{2}\right)L} \mathbb{P}\left(\frac{X_{t}^{(max)}}{\sqrt{4t}}<-L\right) =e^{C_e}. \] These statements are a sharp counterpart of our previous results improved by computing the terms of order $L^{0}$ in the asymptotic expansion of the corresponding Fredholm Pfaffian.

math.PR↗

arXiv2018

Examples of interacting particle systems on $\mathbb{Z}$ as Pfaffian point processes: annihilating and coalescing random walks

A class of interacting particle systems on $\mathbb{Z}$, involving instantaneously annihilating or coalescing nearest neighbour random walks, are shown to be Pfaffan point processes for all deterministic initial conditions. As diffusion limits, explicit Pfaffan kernels are derived for a variety of coalescing and annihilating Brownian systems. For Brownian motions on $\mathbb{R}$, depending on the initial conditions, the corresponding kernels are closely related to the bulk and edge scaling limits of the Pfaffan point process for real eigenvalues for the real Ginibre ensemble of random matrices. For Brownian motions on $\mathbb{R}_{+}$ with absorbing or reflected boundary conditions at zero new interesting Pfaffan kernels appear. We illustrate the utility of the Pfaffan structure by determining the extreme statistics of the rightmost particle for the purely annihilating Brownian motions, and also computing the probability of overcrowded regions for all models.

math.PR↗

arXiv2018

Stationary mass distribution and nonlocality in models of coalescence and shattering

We study the asymptotic properties of the steady state mass distribution for a class of collision kernels in an aggregation-shattering model in the limit of small shattering probabilities. It is shown that the exponents characterizing the large and small mass asymptotic behavior of the mass distribution depend on whether the collision kernel is local (the aggregation mass flux is essentially generated by collisions between particles of similar masses), or non-local (collision between particles of widely different masses give the main contribution to the mass flux). We show that the non-local regime is further divided into two sub-regimes corresponding to weak and strong non-locality. We also observe that at the boundaries between the local and non-local regimes, the mass distribution acquires logarithmic corrections to scaling and calculate these corrections. Exact solutions for special kernels and numerical simulations are used to validate some non-rigorous steps used in the analysis. Our results show that for local kernels, the scaling solutions carry a constant flux of mass due to aggregation, whereas for the non-local case there is a correction to the constant flux exponent. Our results suggest that for general scale-invariant kernels, the universality classes of mass distributions are labeled by two parameters: the homogeneity degree of the kernel and one further number measuring the degree of the non-locality of the kernel.

cond-mat.stat-mech↗

arXiv2017

Multi-point correlations for two dimensional coalescing random walks

This paper considers an infinite system of instantaneously coalescing rate one simple random walks on $\mathbb{Z}^2$, started from the initial condition with all sites in $\mathbb{Z}^2$ occupied. We show that the correlation functions of the model decay, for any $N \geq 2$, as \[ ρ_N (x_1,\ldots,x_N;t) = \frac{c_0(x_1,\ldots,x_N)}{π^N} (\log t)^{N-{N \choose 2}} t^{-N} \left(1 + O\left( \frac{1}{\log^{\frac12-δ}\!t} \right) \right) \] as $t \to\infty$. This generalises the results for $N=1$ due to Bramson and Griffeath and confirms a prediction in the physics literature for $N>1$. An analogous statement holds for instantaneously annihilating random walks. The key tools are the known asymptotic $ρ_1(t) \sim \log t/πt$ due to Bramson and Griffeath, and the non-collision probability $p_{NC}(t)$, that no pair of a finite collection of $N$ two dimensional simple random walks meets by time $t$, whose asymptotic $p_{NC}(t) \sim c_0 (\log t)^{-{N \choose 2}}$ was found by Cox, Merle and Perkins. This paper re-derives the asymptotics both for $ρ_1(t)$ and $p_{NC}(t)$ by proving that these quantities satisfy {\it effective rate equations}, that is approximate differential equations at large times. This approach can be regarded as a generalisation of the Smoluchowski theory of renormalised rate equations to multi-point statistics.

math.PR↗