We study the phase separation phenomenon in the Ising model in dimensions $d \geq 3$. To this end we work in a large box with plus boundary conditions and we condition the system to have an excess amount of negative spins so that the empirical magnetization is smaller than the spontaneous magnetization $m^*$. We confirm the prediction of the phenomenological theory by proving that with high probability a single droplet of the minus phase emerges surrounded by the plus phase. Moreover, the rescaled droplet is asymptotically close to a definite deterministic shape, the Wulff crystal, which minimizes the surface free energy. In the course of the proof we establish a surface order large deviation principle for the magnetization. Our results are valid for temperatures $T$ below a limit of slab-thresholds $\hat{T}_c$ conjectured to agree with the critical point $T_c$. Moreover, $T$ should be such that there exist only two extremal translation invariant Gibbs states at that temperature, a property which can fail for at most countably many values and which is conjectured to be true for every $T$. The proofs are based on the Fortuin–Kasteleyn representation of the Ising model along with coarse-graining techniques.To handle the emerging macroscopic objects we employ tools from geometric measure theory which provide an adequate framework for the large deviation analysis. Finally,we propose a heuristic picture that for subcritical temperatures close enough to $T_c$, the dominant minus spin cluster of the Wulff droplet permeates the entire box and has a strictly positive local density everywhere.

We provide an extremely accurate picture of the Sherrington – Kirkpatrick model in three cases:for high temperature, for large external field and for any temperature greater than or equal to 1 and sufficiently small external field. We describe the system at the level of the central limit theorem, or as physicists would say, at the level of fuctuations around the mean field. We also obtain much more detailed information, in the form of exponential inequalities that express a uniform control over higher order moments.We give a complete, rigorous proof that at the generic point of the predicted low temperature region there is “replica symmetry breaking,” in the sense that the system is unstable with respect to an infinitesimal coupling between two replicas.

Analogues of stepping-stone models are considered where the sitespace is continuous, the migration process is a general Markov process, and the type-space is infinite. Such processes were defined in previous work of the second author by specifying a Feller transition semigroup in terms of expectations of suitable functionals for systems of coalescing Markov processes. An alternative representation is obtained here in terms of a limit of interacting particle systems. It is shown that, under a mild condition on the migration process, the continuum-sites stepping-stone process has continuous sample paths. The case when the migration process is Brownian motion on the circle is examined in detail using a duality relation between coalescing and annihilating Brownian motion. This duality relation is also used to show that a tree-like random compact metric space that is naturally associated to an in .nite family of coalescing Brownian motions on the circle has Hausdorff and packing dimension both almost surely equal to 1/2 and, moreover, this space is capacity equivalent to the middle-1/2 Cantor set (and hence also to the Brownian zero set).

Say that a graph has persistent transition if the Ising model on the graph can exhibit a phase transition (nonuniqueness of Gibbs measures) in the presence of a nonzero external field.We show that for nonamenable graphs, for Bernoulli percolation with $p$ close to 1, all the infinite clusters have persistent transition.On the other hand, we show that for transitive amenable graphs, the infinite clusters for any stationary percolation do not have persistent transition. This extends a result of Georgii for the cubic lattice. A geometric consequence of this latter fact is that the infinite clusters are strongly amenable (i.e., their anchored Cheeger constant is 0). Finally we show that the critical temperature for the Ising model with no external field on the infinite clusters of Bernoulli percolation with parameter $p$, on an arbitrary bounded degree graph, is a continuous function of $p$.

We study bounded $p$-harmonic functions $u$ defined on a directed tree $T$ with branching order $\kappa(1<p<\infty$ \and $\kappa=2,3,\ldots)$. Denote by $BV(u)$ the set of paths on which $u$ has finite variation and $\mathscr{F}(u)$ the set of paths on which $u$ has a finite limit. Then the infimum of dim $BV(u)$ and the infimum of dim $\mathscr{F}(u)$ are equal over all bounded-harmonic functions on $T$ (with $p$ and $\kappa$ fixed); the infimum $d(\kappa, p)$ is attained and is strictly between 0 and 1 expect when $p = 2$ or $\kappa = 2$.

We consider interacting particle systems on $Z$ which allow five types of pairwise interaction: Annihilation, Birth, Coalescence, Death and Exclusion with corresponding rates a, b,c, d, e . We show that whatever the values of a, c, d, e, if the birthrate is high enough there is a positive probability the particle system will survive starting from any finite occupied set. In particular: an IPS with rates a b c d e has a positive probability of survival if

$$ b > 4d + 6a, \quad c + a \geq d + e$$

or

$$ b > 7d + 3a - 3c + 3e, \quad c + a < d + e.$$

We create a suitable supermartingale by extending the method used by Holley and Liggett in their treatment of the contact process.

We study the .nite zero-range process with occupancy-dependent rate function $g(\cdot)$. Under the invariant measure, which can be written explicitly in terms of $g$, particles are distributed over sites and we regard all particles at a fixed site as a cluster. In the density one case, that is, equal numbers of particles and sites, we determine asymptotically the size of the largest cluster, as the number of particles tends to infinity, and determine its dependence on the rate function.

For a series of randomly discounted terms we give an integral criterion to distinguishbetween almost-sure absolute convergence and divergence in probability to $\infty$, these being the only possible forms of asymptotic behavior. This solves the existence problem for a one-dimensional perpetuity that remains from a 1979 study by Vervaat, and yields a complete characterization of the existence of distributional fixed points of a random affine map in dimension one.

We consider an in .nite Galton–Watson tree $\Gamma$ and label the vertices $v$ with a collection of i.i.d.random variables $(Y_v)_{v \in \Gamma}$. In the case where the upper tail of the distribution of $Y_v$ is semiexponential, we then determine the speed of the corresponding tree-indexed random walk. In contrast to the classical case where the random variables $Y_v$ have finite exponential moments, the normalization in the definition of the speed depends on the distribution of $Y_v$. Interpreting the random variables $Y_v$ as displacements of the offspring from the parent, $(Y_v)_{v \in \Gamma}$ describes a branching random walk. The result on the speed gives a limit theorem for the maximum of the branching random walk, that is, for the position of the rightmost particle. In our case, this maximum grows faster than linear in time.

We present a method for proving Perron –Frobenius eigenvalue inequalities. The method is to apply Jensen’s inequality to the change in a “random evolution ”over a regenerative cycle of the underlying finite-state Markov chain. One of the primary benefits of the method is that it readily gives necessary and sufficient conditions for strict inequality. It also gives insights into some of the conjectures of J.E.Cohen. Ney and Nummelin’s “Hypothesis 2” arises here as a condition for strict inequality, and we explore its ramifications in detail for a special family of Markov additive processes which we call “fluid models.” This leads to a connection between Hypothesis 2 and the condition “$P^T P$ irreducible” which arose in the work of Cohen, Friedland, Kato and Kelly.

The existence of a dominating point for an open convex set and a corresponding representation formula for large deviation probabilities are established in the infinite-dimensional setting under conditions which are both necessary and sufficient and follow from those used previously in $\mathbb{R}^d$ . A precise nonlogarithmic estimate of large deviation probabilities applicable to Gaussian measures is also included.

The standard normal distribution $\Phi$ on $\mathbb{R}^d$ satisfies $\Phi((\partial C)^\varepsilon)\leq c_d \varepsilon$, for all $\varepsilon > 0$ and for all convex subsets $C \subset \mathbb{R}^d$, with a constant $c_d$ which depends on the dimension $d$ only. Here $\partial C$ denotes the boundary of $C$, and $(\partial C)^\epsilon$ stands for the $\epsilon$-neighborhood of $\partial C$. Such bounds for the normal measure of convex shells are extensively used to estimate the accuracy of normal approximations.

We extend the inequality to all (nondegenerate) stable distributions on $\mathbb{R}^d$, with a constant which depends on the dimension, the characteristic exponent and the spectral measure of the distribution only. As a corollary we provide an explicit bound for the accuracy of stable approximations on the class of all convex subsets of $\mathbb{R}^d$ .

We prove functional limit laws for the increment functions of empirical processes based upon randomly right-censored data. The increment sizes we consider are classified into different classes covering the whole possible spectrum. We apply these results to obtain a description of the strong limiting behavior of a series of estimators of local functionals of lifetime distributions. In particular, we treat the case of kernel density and hazard rate estimators.

We prove a central limit theorem for strongly mixing sequences under a sharp sufficient condition which combines the rate of the strong mixing coefficient with the quantile function. The result improves on all earlier central limit theorems for this type of dependence and answers a conjecture raised by Bradley in 1997.

Moreover, we derive the corresponding functional central limit theorem.

We discuss the CLT for the global and local linear statistics of random matrices from classical compact groups. The mainparts of our proofs are certain combinatorial identities, much in the spirit of works by M. Kac and H. Spohn.

Matsumoto and Yor have recently discovered an interesting transformation which preserves a bivariate probability measure which is a product of the generalized inverse Gaussian (GIG) and gamma distributions. This paper is devoted to a detailed study of this phenomenon. Let $X$ and $Y$ be two independent positive random variables. We prove (Theorem 4.1) that $U =(X +Y)^{-1}$ and $V = X^{-1} - (X +Y)^{-1}$ are independent if and only if there exists $p, a, b > 0$ such that $Y$ is gamma distributed with shape parameter $p$ and scale parameter $2 a^-1$, and such that $X$ has a GIG distribution with parameters $-p, a$ and $b$ (the direct part for $a = b$ was obtained in Matsumoto and Yor). The result is partially extended (Theorem 5.1) to the case where $X$ and $Y$ are valued in the cone $V_+$ of symmetric positive definite $(r, r)$ real matrices as follows: under a hypothesis of smoothness of densities, we prove that $U =(X +Y)^-1$ and $V =X^-1 -(X +Y)^ -1$ are independent if and only if there exists $p>(r-1)/2$ and $a$ and $b$ in $V_+$ such that $Y$ is Wishart distributed with shape parameter $p$ and scale parameter $2a^-1$, and such that $X$ has a matrix GIG distribution with parameters $-p, a$ and $b$. The direct result is also extended to singular Wishart distributions (Theorem 3.1).

The objective of this paper is to present two theorems which are directly applicable to optimal stopping problems involving independent indicator functions. The proofs are elementary. One implication of the results is a convenient solution algorithm to obtain the optimal stopping rule and the value.We will apply it to several examples of sequences of independent indicators, including sequences of random length. Another interesting implication of the results is that the well-known asymptotic value $1 / e$ for the classical best-choice problem is in fact a typical lower boundin a much more general class of problems.