Interacting particle systems is by now a mature area of probability theory, but one that is still very active. We begin this paper by explaining how models from this area arise in fields such as physics and biology. We turn then to a discussion of both older and more recent results about them, concentrating on contact processes, voter models, and exclusion processes. These processes are among the most studied in the field, and have the virtue of relative simplicity in their description, which permits us to address the fundamental issues about their behavior without dealing with the extra complications that models from specific areas of application would require.

For a nonnegative subadditive function $h$ on $\mathbb{Z}^d$, with limiting approximation $g(x) = \lim_n h(nx)/n$, it is of interest to obtain bounds on the discrepancy between $g(x)$ and $h(x)$, typically of order $|x|^{\nu}$ with $\nu < 1$. For certain subadditive $h(x)$, particularly those which are expectations associated with optimal random paths from 0 to $x$, in a somewhat standardized way a more natural and seemingly weaker property can be established: every $x$ is in a bounded multiple of the convex hull of the set of sites satisfying a similar bound. We show that this convex-hull property implies the desired bound for all $x$. Applications include rates of convergence in limiting-shape results for first-passage percolation (standard and oriented) and longest common subsequences and bounds on the error in the exponential-decay approximation to the off-axis connectivity function for subcritical Bernoulli bond percolation on the integer lattice.

We consider two critical spatial branching processes on $\mathbb{R}^d$: critical branching Brownian motion, and the critical Dawson-Watanabe process. A basic feature of these processes is that their ergodic behavior is highly dimension dependent. It is known that in low dimensions, $d \leq 2$, the only invariant measure is $\delta_0$ , the unit point mass on the empty state. In high dimensions, $d \geq 3$, there is a family ${\nu_{\theta}, \theta \epsilon [0, \infty)}$ of extremal invariant measures; the measures ${\nu_{\theta}$ are translation invariant and indexed by spatial intensity. We prove here, for $d \geq 3$, that all invariant measures are convex combinations of these measures.

4 We consider families of {0, 1}-valued random variables indexed by the vertices of countable graphs with bounded degree. First we show that if these random variables satisfy the property that conditioned on what happens outside of the neighborhood of each given site, the probability of seeing a 1 at this site is at least a value $p$ which is large enough, then this random field dominates a product measure with positive density. Moreover the density of this dominated product measure can be made arbitrarily close to 1, provided that $p$ is close enough to 1. Next we address the issue of obtaining the critical value of $p$, defined as the threshold above which the domination by positive-density product measures is assured. For the graphs which have as vertices the integers and edges connecting vertices which are separated by no more than $k$ units, this critical value is shown to be $1 - k^k /(k + 1)^{k+1}$, and a discontinuous transition is shown to occur. Similar critical values of $p$ are found for other classes of probability measures on ${0, 1}^{\mathbb{Z}}$. For the class of $k$-dependent measures the critical value is again $1 - k^k /(k + 1)^{k+1}$, with a discontinuous transition. For the class of two-block factors the critical value is shown to be 1/2 and a continuous transition is shown to take place in this case. Thus both the critical value and the nature of the transition are different in the two-block factor and 1-dependent cases.

Duality has proved to be a powerful technique in the study of interacting particle systems (IPS). This concept can be enlarged and a “quasi-duality” defined between various pairs of IPS previously thought unrelated. Consequently, theorems of a similar style to those involving duality can be deduced.

The concept of quasi-duality follows naturally from our previous studies into the use of “single-site operators” (an idea borrowed from quantum physics) in paper II of this series. It is shown that a necessary condition for quasi-duality is that the eigenvalues of the corresponding two-site infinitesimal generators be the same, and, using this observation, a number of quasi-dual pairs have been found and studied.

It is further shown that if two different IPS share a common dual, then one can be considered as a “thinning” of the other.

We specify the Markov process corresponding to a generalized mollified Boltzmann equation with general motion between collisions and nonlinear bounded jump (collision) operator, and give the nonlinear martingale problem it solves. We consider various linear interacting particle systems in order to approximate this nonlinear process. We prove propagation of chaos, in variation norm on path space with a precise rate of convergence, using coupling and interaction graph techniques and a representation of the nonlinear process on a Boltzmann tree. No regularity nor uniqueness assumption is needed. We then consider a nonlinear equation with both Vlasov and Boltzmann terms and give a weak pathwise propagation of chaos result using a compactness-uniqueness method which necessitates some regularity. These results imply functional laws of large numbers and extend to multitype models. We give algorithms simulating or approximating the particle systems.

.1 2 We consider the equation $u_{tt} = \Delta u + a(u) \mathsf{N}$ for $x \epsilon \mathbf{R}^1$ or $R^2$. $\mathsf{N}$ is a Gaussian noise term, which is white noise if $x \epsilon \mathbf{R}^1$. If $a(u)$ grows no faster than $u (\log u)^{1/2-\varepsilon}$, then there is a unique solution valid for all time.

Let $X^n = (X_t^n, \mathscr{F}_t^n)_{0 \leq t \leq 1}$ be square integrable martingales with the quadratic characteristics $\langle X^n \rangle, n = 1, 2, \dots$. We prove that the large deviations relation $P(X_1^n \geq r)/(1 - \Phi (r)) \to 1$ holds true for $r$ growing to infinity with some rate depending on $L_{2\delta}^n = E \sum_{0\leq t\leq 1}| \Delta X_t^n |^{2 + 2 \delta}$ and $N_{2 \delta}^n = E | \langle X^n \rangle_1 - 1|^{1 + \delta}$, where $\delta > 0$ and $L_{2 \delta}^n \to 0$, N_{2 \delta}^n \to 0$ as $n \to \infty$. The exact bound for the remainder is also obtained.

A dimension free lower bound is found for isoperimetric constants of product probability measures. From this, some analytic inequalities are derived.

We prove an isoperimetric inequality on the discrete cube which is the precise analog of a logarithmic inequality due to Talagrand. As a consequence, the Gaussian isoperimetric inequality is derived.

A random variable $X$ is C-decomposable if $X =_D cX + Y_c$ for all $c$ in $C$, where $_c$ is a random variable independent of $X$ and $C$ is a closed multiplicative subsemigroup of [0, 1]. $X$ is self-decomposable if $C = [0, 1]$ . Extending an idea of Urbanik in the self-decomposable case, we define a decreasing sequence of subclasses of the class of $C$-decomposable laws, for any $C$. We give a structural representation for laws in these classes, and we show that laws in the limiting subclass are infinitely divisible. We also construct noninfinitely divisible examples, some of which are continuous singular.

Let $X$ and $Y$ be symmetric diffusion processes with a common state space, and let $P^m$ (resp.$Q^{\mu}$) be the law of $X$ (resp. $Y$) with its symmetry measure $m$ (resp.$\mu$) as initial distribution. We study the consequences of the absolute continuity condition $Q^{\mu} \ll_{\loc} P^m$. We show that under this condition there is a "smooth" version $\rho$ of the Radon-Nikodym derivative $d \mu/dm$ such that $1/2[\log \rho(X_t) - \log \rho(X_0)] = M_t + N_t, t < \sigma$, where $M$ is a continuous local martingale additive functional, $N$ is a zero-energy continuous additive functional and $\sigma$ is an explosion time. The Girsanov density $L_t := dQ^{\mu} |_{F_t}/dP^m|_{F_t}$ then admits the representation $L_t = \exp(M_t - 1/2 \langle M \rangle_t)1_{{t \leq \sigma}}$. The density $\rho$ also serves to link the Dirichlet forms of $X$ and $Y$ in a simple way. Our identification of $L$ relies on notions of even and oddfor additive functionals. These notions complement Fukushima’s decomposition and the forward-backward martingale decomposition of Lyons and Zheng.

We obtain lower bounds for moderate deviations of empirical measures of a Markov chain with general state space under the assumption of ergodicity of degree 2. We derive an explicit expression for the rate function.

Let ${X, X_n, n \geq 1}$ be a sequence of independent and identically distributed random variables. The classical Cramér-Chernoff large deviation states that $\lim_{n\to\infty} n^{-1} \ln P((\sum_{i=1}^n X_i)/n \geq x) = \ln \rho (x)$ if and only if the moment generating function of $X$ is finite in a right neighborhood of zero. This paper uses $n^{(p-1)/p} V_{n,p} = n^{(p-1)/p}(\sum_{i=1}^n |X_i|^p)^{1/p} (p > 1)$ as the normalizing constant to establish a self-normalized large deviation without any moment conditions. A self-normalized moderate deviation, that is, the asymptotic probability of $P(S_n/V_{n,p} \geq x_n) for $x_n = o(n^{(p-1)/p})$, is also found for any $X$ in the domain of attraction of a normal or stable law. As a consequence, a precise constant in the self-normalized law of the iterated logarithm of Griffin and Kuelbs is obtained. Applications to the limit distribution of self-normalized sums, the asymptotic probability of the $t$-statistic as well as to the Erdös-Rényi-Shepp law of large numbers are also discussed.

It is known that the length $L(x_1^n)$ of the longest block appearing at least twice in a randomly chosen sample path of length $n$ drawn from an i.i.d. process is asymptotically almost surely equal to $C \log n$, where the constant $C$ depends on the process. A simple coding argument will be used to show that for a class of processes called the finite energy processes, $L(x_1^n)$ is almost surely upper bounded by $C \log n$, where $C$ is a constant. While the coding technique does not yield the exact constant $C$, it does show clearly what is needed to obtain log $n$ bounds.

In the discrete-time supercritical branching random walk, there is a Kesten-Stigum type result for the martingales formed by the Laplace transform of the $n$th generation positions. Roughly, this says that for suitable values of the argument of the Laplace transform the martingales converge in mean provided an "$X \log X$" condition holds. Here it is established that when this moment condition fails, so that the martingale ..converges to zero, it is possible to find a (Seneta-Heyde) renormalization of the martingale that converges in probability to a finite nonzero limit when the process survives. As part of the proof, a Seneta-Heyde renormalization of the general (Crump-Mode-Jagers) branching process is obtained; in this case the convergence holds almost surely. The results rely heavily on a detailed study of the functional equation that the Laplace transform of the limit must satisfy.

We discuss and disprove a conjecture of Pemantle concerning vertex-reinforced random walks.

The setting is a general theory of non-Markovian discrete-time random 4 processes on a finite space $E = {1, \dots, d}$, for which the transition probabilities at each step are influenced by the proportion of times each state has been visited. It is shown that, under mild conditions, the asymptotic behavior of the empirical occupation measure of the process is precisely related to the asymptotic behavior of some deterministic dynamical system induced by a vector field on the $d - 1$ unit simplex. In particular, any minimal attractor of this vector field has a positive probability to be the limit set of the sequence of empirical occupation measures. These properties are used to disprove a conjecture and to extend some results due to Pemantle. Some applications to edge-reinforced random walks are also considered.

In relation with Monte Carlo methods to solve some integro-differential equations, we study the approximation problem of $\mathbb{E}g(X_T)$ by $\mathbb{E}g(\overline{X}_T^n)$, where $(X_t, 0 \leq t \leq T)$ is the solution of a stochastic differential equation governed by a Lévy process $(Z_t), (\overline{X}_t^n)$ is defined by the Euler discretization scheme with step $T/n$. With appropriate assumptions on $g(\cdot)$, we show that the error of $\mathbb{E}g(X_T) - \mathbb{E}g(\overline{X}_T^n)$ can be expanded in powers of $1/n$ if the Lévy measure of $Z$ has finite moments of order high enough. Otherwise the rate of convergence is slower and its speed depends on the behavior of the tails of the Lévy measure.

The behavior of tail probabilities $\mathbf{P}{S \leq r}, r \to 0$ is investigated, where $S$ is a series $S = \Sigma \lambda_j Z_j$ generated by some sequence of positive numbers ${\lambda_j}$ and by a sequence ${Z_j}$ of independent copies of a positive random variable $Z$.

We present the exact asymptotic expression for $\mathbf{P}{S \leq r}$ by means of Laplace transform $\Lambda (\gamma) = \mathbf{E} \exp {- \gamma S}$ under weak assumptions on the behavior of the tail probabilities of $Z$ in the vicinity of zero. The bounds of accuracy are also given, and under weak supplementary smoothness conditions the asymptotic properties of the density of $S$ are investigated.

In this paper we study the CLT for partial sums of a generalized linear process $X_n = \sum_{i=1}^n a_{ni} \xi_i$, where $\sup_n \sum_{i=1}^n a_{ni}^2 < \infty, \max_{1 \leq i \leq n}are in turn, pairwise mixing martingale differences, mixing sequences or associated sequences. The results are important in analyzing the asymptotical properties of some estimators as well as of linear processes.

In this paper we prove central limit theorems of the following kind: let $S^d \subset \mathbb{R}^{d + 1}$ be the unit sphere of dimension $d \geq 2$ with uniform distribution $\omega_d$. For each $k \epsilon \mathbb{N}$, consider the isotropic random walk $(X_n^k)_{n \geq 0}$ on $S^d$ starting at the north pole with jumps of fixed sizes $\angle (X_n^k, X_{n - 1}^k) = \pi/\sqrt{k}$ for all $n \geq 1$. Then there is some $k_0(d)$ such that for all $k \geq k_0(d)$, the distributions $\varrho_k$ of $X_k^k$ have continuous, bounded $\omega_d$-densities $f_k$. Moreover, there is a (known) Gaussian measure $\nu$ on $S^d$ with $\omega_d$-density such that $||f_k - h||_{\infty} = O(1/k)$ and $||\varrho_k - \nu|| = O(1/k)$ for $k \to \infty$, where $O(1/k)$ is sharp. We shall derive this rate of convergence in the central limit theorem more generally for a quite general class of isotropic random walks on compact symmetric spaces of rank one as well as for random walks on $[0, \pi]$ whose transition probabilities are related to product linearization formulas of Jacobi polynomials.

Let $(B_n, A_n)_{n \geq 1}$ be a sequence of i.i.d. random variables with values in $\mathbf{R}^d \times \mathbf{R}_*^+$. The Markov chain on $\mathbf{R}^d$ which satisfies the random equa tion $X_n = A_n X_{n-1} + B_n$ is studied when $\mathbf{E}(\log A_1) = 0$. No density assumption on the distribution of $(B_1, A_1)$ is made. The main results are recurrence of the Markov chain $X_n$, stability properties of the paths, existence and uniqueness of a Radon invariant measure and a limit theorem for the occupation times. The results rely on a renewal theorem for the process $(X_n, A_n \dots A_1)$.

It was conjectured in the early 1950’s that the empirical spectral distribution of an $n \times n$ matrix, of iid entries, normalized by a factor of $\frac{1}{\sqrt{n}}$, converges to the uniform distribution over the unit disc on the complex plane, which is called the circular law. Only a special case of the conjecture, where the entries of the matrix are standard complex Gaussian, is known. In this paper, this conjecture is proved under the existence of the sixth moment and some smoothness conditions. Some extensions and discussions are also presented.