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 $$\rho_N (x_1,\ldots,x_N;t) = \frac{c_0(x_1,\ldots,x_N)}{\pi^N} (\log t)^{N-{N \choose 2}} t^{-N} \left(1 + O\left( \frac{1}{\log^{\frac12-\delta}\!t} \right) \right)$$ 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 $\rho_1(t) \sim \log t/\pi t$ 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 $\rho_1(t)$ 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.Comment: 26 page

Pfaffian formulae for one dimensional coalescing and annihilating systems

The paper considers instantly coalescing, or instantly annihilating, systems of one-dimensional Brownian particles on the real line. Under maximal entrance laws, the distribution of the particles at a fixed time is shown to be Pfaffian point processes closely related to the Pfaffian point process describing one dimensional distribution of real eigenvalues in the real Ginibre ensemble of random matrices. As an application, an exact large time asymptotic for the n-point density function for coalescing particles is derived

On the threshold of spread-out voter model percolation

In the $R$-spread out, $d$-dimensional voter model, each site $x$ of $\mathbb{Z}^d$ has state (or `opinion') 0 or 1 and, with rate 1, updates its opinion by copying that of some site $y$ chosen uniformly at random among all sites within distance $R$ from $x$. If $d \geq 3$, the set of (extremal) stationary measures of this model is given by a family $\mu_{\alpha, R}$, where $\alpha \in [0,1]$. Configurations sampled from this measure are polynomially correlated fields of 0's and 1's in which the density of 1's is $\alpha$ and the correlation weakens as $R$ becomes larger. We study these configurations from the point of view of nearest neighbor site percolation on $\mathbb{Z}^d$, focusing on asymptotics as $R \to \infty$. In [Ráth, Valesin, AoP, 2017] we have shown that, if $R$ is large, there is a critical value $\alpha_c(R)$ such that there is percolation if $\alpha > \alpha_c(R)$ and no percolation if $\alpha < \alpha_c(R)$. Here we prove that, as $R \to \infty$, $\alpha_c(R)$ converges to the critical probability for Bernoulli site percolation on $\mathbb{Z}^d$. Our proof relies on a new upper bound on the joint occurrence of events under $\mu_{\alpha,R}$ which is of independent interest

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.Comment: 62 pages, 2 figure

Universality classes in nonequilibrium lattice systems

This work is designed to overview our present knowledge about universality classes occurring in nonequilibrium systems defined on regular lattices. In the first section I summarize the most important critical exponents, relations and the field theoretical formalism used in the text. In the second section I briefly address the question of scaling behavior at first order phase transitions. In section three I review dynamical extensions of basic static classes, show the effect of mixing dynamics and the percolation behavior. The main body of this work is given in section four where genuine, dynamical universality classes specific to nonequilibrium systems are introduced. In section five I continue overviewing such nonequilibrium classes but in coupled, multi-component systems. Most of the known nonequilibrium transition classes are explored in low dimensions between active and absorbing states of reaction-diffusion type of systems. However by mapping they can be related to universal behavior of interface growth models, which I overview in section six. Finally in section seven I summarize families of absorbing state system classes, mean-field classes and give an outlook for further directions of research.Comment: Updated comprehensive review, 62 pages (two column), 29 figs included. Scheduled for publication in Reviews of Modern Physics in April 200

Universality classes in nonequilibrium lattice systems

This work is designed to overview our present knowledge about universality classes occurring in nonequilibrium systems defined on regular lattices. In the first section I summarize the most important critical exponents, relations and the field theoretical formalism used in the text. In the second section I briefly address the question of scaling behavior at first order phase transitions. In section three I review dynamical extensions of basic static classes, show the effect of mixing dynamics and the percolation behavior. The main body of this work is given in section four where genuine, dynamical universality classes specific to nonequilibrium systems are introduced. In section five I continue overviewing such nonequilibrium classes but in coupled, multi-component systems. Most of the known nonequilibrium transition classes are explored in low dimensions between active and absorbing states of reaction-diffusion type of systems. However by mapping they can be related to universal behavior of interface growth models, which I overview in section six. Finally in section seven I summarize families of absorbing state system classes, mean-field classes and give an outlook for further directions of research.Comment: Updated comprehensive review, 62 pages (two column), 29 figs included. Scheduled for publication in Reviews of Modern Physics in April 200

Multi-Scaling of Correlation Functions in Single Species Reaction-Diffusion Systems

We derive the multi-fractal scaling of probability distributions of multi-particle configurations for the binary reaction-diffusion system $A+A \to \emptyset$ in $d \leq 2$ and for the ternary system $3A \to \emptyset$ in $d=1$. For the binary reaction we find that the probability $P_{t}(N, \Delta V)$ of finding $N$ particles in a fixed volume element $\Delta V$ at time $t$ decays in the limit of large time as $(\frac{\ln t}{t})^{N}(\ln t)^{-\frac{N(N-1)}{2}}$ for $d=2$ and t^{-Nd/2}t^{-\frac{N(N-1)\epsilon}{4}+\mathcal{O}(\ep^2)} for $d<2$. Here \ep=2-d. For the ternary reaction in one dimension we find that $P_{t}(N,\Delta V) \sim (\frac{\ln t}{t})^{N/2}(\ln t)^{-\frac{N(N-1)(N-2)}{6}}$. The principal tool of our study is the dynamical renormalization group. We compare predictions of \ep-expansions for $P_{t}(N,\Delta V)$ for binary reaction in one dimension against exact known results. We conclude that the \ep-corrections of order two and higher are absent in the above answer for $P_{t}(N, \Delta V)$ for $N=1,2,3,4$. Furthermore we conjecture the absence of \ep^2-corrections for all values of $N$.Comment: 10 pages, 6 figure

One-dimensional interacting particle systems as Pfaffian point processes

A wide class of one-dimensional continuous-time discrete-space interacting particle systems are shown to be Pfaffian point processes at fixed times with kernels characterised by the solutions to associated two-dimensional ODEs. The models comprise instantaneously coalescing or annihilating random walks with fully spatially inhomogeneous jump rates and deterministic initial conditions, including additional pairwise immigration or branching in the pure interaction regimes. We formulate convergence of Pfaffian point processes via their kernels, enabling investigation of diffusive scaling limits, which boils down uniform convergence of lattice approximations to two-dimensional PDEs. Convergence to continuum point processes is developed for a subset of the discrete models. Finally, in the case of annihilating random walks with pairwise immigration we extend the picture to multiple times, establishing the extended Pfaffian property for the temporal process

A Course in Interacting Particle Systems

These lecture notes give an introduction to the theory of interacting particle systems. The main subjects are the construction using generators and graphical representations, the mean field limit, stochastic order, duality, and the relation to oriented percolation. An attempt is made to give a large number of examples beyond the classical voter, contact and Ising processes and to illustrate these based on numerical simulations.Comment: These are lecture notes for a course in interacting particle systems taught at Charles University, Prague, in 2015/2016 and again in the fall of 2019. Compared to the first version, a number of small typos and mistakes have been corrected, most notably the proof of Lemma 4.18, which was wrong in the first version. Some parts have been rephrased for greater clarit