We solve an open problem concerning the relaxation time (inverse spectral gap) of the zero range process in Z^d/LZ^d with constant rate, proving a tight upper bound of O((ρ+1)²L²), where ρ is the density of particles.

Minimal spanning forests on infinite graphs are weak limits of minimal spanning trees from finite subgraphs. These limits can be taken with free or wired boundary conditions and are denoted FMSF (free minimal spanning forest) and WMSF (wired minimal spanning forest), respectively. The WMSF is also the union of the trees that arise from invasion percolation started at all vertices. We show that on any Cayley graph where critical percolation has no infinite clusters, all the component trees in the WMSF have one end a.s. In ℤ^d this was proved by Alexander [Ann. Probab. 23 (1995) 87–104], but a different method is needed for the nonamenable case. We also prove that the WMSF components are “thin” in a different sense, namely, on any graph, each component tree in the WMSF has p_c=1 a.s., where p_c denotes the critical probability for having an infinite cluster in Bernoulli percolation. On the other hand, the FMSF is shown to be “thick”: on any connected graph, the union of the FMSF and independent Bernoulli percolation (with arbitrarily small parameter) is a.s. connected. In conjunction with a recent result of Gaboriau, this implies that in any Cayley graph, the expected degree of the FMSF is at least the expected degree of the FSF (the weak limit of uniform spanning trees). We also show that the number of infinite clusters for Bernoulli(p_u) percolation is at most the number of components of the FMSF, where p_u denotes the critical probability for having a unique infinite cluster. Finally, an example is given to show that the minimal spanning tree measure does not have negative associations.

We prove a general noncommutative law of large numbers. This applies in particular to random walks on any locally finite homogeneous graph, as well as to Brownian motion on Riemannian manifolds which admit a compact quotient. It also generalizes Oseledec’s multiplicative ergodic theorem. In addition, we show that ɛ-shadows of any ballistic random walk with finite moment on any group eventually intersect. Some related results concerning Coxeter groups and mapping class groups are recorded in the last section.

Threshold phenomena are investigated using a general approach, following Talagrand [Ann. Probab. 22 (1994) 1576–1587] and Friedgut and Kalai [Proc. Amer. Math. Soc. 12 (1999) 1017–1054]. The general upper bound for the threshold width of symmetric monotone properties is improved. This follows from a new lower bound on the maximal influence of a variable on a Boolean function. The method of proof is based on a well-known logarithmic Sobolev inequality on {0,1}ⁿ. This new bound is shown to be asymptotically optimal.

The influence theorem for product measures on the discrete space {0,1}^N may be extended to probability measures with the property of monotonicity (which is equivalent to “strong positive association”). Corresponding results are valid for probability measures on the cube [0,1]^N that are absolutely continuous with respect to Lebesgue measure. These results lead to a sharp-threshold theorem for measures of random-cluster type, and this may be applied to box crossings in the two-dimensional random-cluster model.

In this paper we consider directed polymers in random environment with discrete space and time. For transverse dimension at least equal to 3, we prove that diffusivity holds for the path in the full weak disorder region, that is, where the partition function differs from its annealed value only by a nonvanishing factor. Deep inside this region, we also show that the quenched averaged energy has fluctuations of order 1. In complete generality (arbitrary dimension and temperature), we prove monotonicity of the phase diagram in the temperature.

Let (T,d) be a metric space and φ:ℝ₊→ℝ an increasing, convex function with φ(0)=0. We prove that if m is a probability measure m on T which is majorizing with respect to d, φ, that is, $\mathscr{S}:=\sup_{x\in T}\int^{D(T)}_{0}\varphi^{-1}(\frac {1}{m(B(x,\varepsilon ))})\,d\varepsilon <\infty$, then $$\mathbf{E} \sup_{s,t \in T}|X(s)−X(t)|\le32\mathscr{S}$$ for each separable stochastic process X(t), t∈T, which satisfies $\mathbf{E}\varphi(\frac {|X(s)-X(t)|}{d(s,t)})\leq 1$ for all s, t∈T, s≠t. This is a strengthening of one of the main results from Talagrand [Ann. Probab. 18 (1990) 1–49], and its proof is significantly simpler.

We introduce a new family of distributions to approximate ℙ(W∈A) for A⊂{…, −2, −1, 0, 1, 2, …} and W a sum of independent integer-valued random variables ξ₁, ξ₂, …, ξ_n with finite second moments, where, with large probability, W is not concentrated on a lattice of span greater than 1. The well-known Berry–Esseen theorem states that, for Z a normal random variable with mean $\mathbb {E}(W)$ and variance Var (W), ℙ(Z∈A) provides a good approximation to ℙ(W∈A) for A of the form (−∞, x]. However, for more general A, such as the set of all even numbers, the normal approximation becomes unsatisfactory and it is desirable to have an appropriate discrete, nonnormal distribution which approximates W in total variation, and a discrete version of the Berry–Esseen theorem to bound the error. In this paper, using the concept of zero biasing for discrete random variables (cf. Goldstein and Reinert [J. Theoret. Probab. 18 (2005) 237–260]), we introduce a new family of discrete distributions and provide a discrete version of the Berry–Esseen theorem showing how members of the family approximate the distribution of a sum W of integer-valued variables in total variation.

We study the characteristic function and moments of the integer-valued random variable ⌊X+α⌋, where X is a continuous random variables. The results can be regarded as exact versions of Sheppard’s correction. Rounded variables of this type often occur as subsequence limits of sequences of integer-valued random variables. This leads to oscillatory terms in asymptotics for these variables, something that has often been observed, for example in the analysis of several algorithms. We give some examples, including applications to tries, digital search trees and Patricia tries.

Brooks and Makover introduced an approach to studying the global geometric quantities (in particular, the first eigenvalue of the Laplacian, injectivity radius and diameter) of a “typical” compact Riemann surface of large genus based on compactifying finite-area Riemann surfaces associated with random cubic graphs; by a theorem of Belyi, these are “dense” in the space of compact Riemann surfaces. The question as to how these surfaces are distributed in the Teichmüller spaces depends on the study of oriented cycles in random cubic graphs with random orientation; Brooks and Makover conjectured that asymptotically normalized cycle lengths follow Poisson–Dirichlet distribution. We present a proof of this conjecture using representation theory of the symmetric group.

We consider Feller processes on a complete separable metric space X satisfying the ergodic condition of the form $$\mathop{\lim\sup}_{n\rightarrow\infty}\Biggl(\frac{1}{n}\sum_{i=1}^{n}P^{i}(x,O)\Biggr)>0\qquad\mbox{for some }x\in X,$$ where O is an arbitrary open neighborhood of some point z∈X and P is a transition function. It is shown that e-chains which satisfy the above condition admit an invariant probability measure. Some results on the stability of such processes are also presented.

We study a class of stochastic control problems where a cost of the form $$\mathbb{E}\int_{[0,\infty)}e^{-\beta s}[\ell(X_{s})\,ds+h(Y^{\circ}_{s})\,d|Y|_{s}]$$ is to be minimized over control processes Y whose increments take values in a cone $\mathbb{Y}$ of ℝ^p, keeping the state process X=x+B+GY in a cone $\mathbb{X}$ of ℝ^k, k≤p. Here, $x\in\mathbb{X}$, B is a Brownian motion with drift b and covariance Σ, G is a fixed matrix, and Y^○ is the Radon–Nikodym derivative dY/d|Y|. Let ℒ=−(1/2)trace(ΣD²)−b⋅D where D denotes the gradient. Solutions to the corresponding dynamic programming PDE, $$[(\mathscr{L}+\beta)f-\ell]\vee\sup_{y\in\mathbb{Y}:|Gy|=1}[-Gy\cdot Df-h(y)]=0,$$ on $\mathbb{X}^{o}$ are considered with a polynomial growth condition and are required to be supersolution up to the boundary (corresponding to a “state constraint” boundary condition on $\partial\mathbb{X}$). Under suitable conditions on the problem data, including continuity and nonnegativity of ℓ and h, and polynomial growth of ℓ, our main result is the unique viscosity-sense solvability of the PDE by the control problem’s value function in appropriate classes of functions. In some cases where uniqueness generally fails to hold in the class of functions that grow at most polynomially (e.g., when h=0), our methods provide uniqueness within the class of functions that, in addition, have compact level sets. The results are new even in the following special cases: (1) The one-dimensional case k=p=1, $\mathbb{X}=\mathbb{Y}=\mathbb{R}_{+}$; (2) The first-order case Σ=0; (3) The case where ℓ and h are linear. The proofs combine probabilistic arguments and viscosity solution methods. Our framework covers a wide range of diffusion control problems that arise from queueing networks in heavy traffic.

We consider the existence and pathwise uniqueness of the stochastic heat equation with a multiplicative colored noise term on ℝ^d for d≥1. We focus on the case of non-Lipschitz noise coefficients and singular spatial noise correlations. In the course of the proof a new result on Hölder continuity of the solutions near zero is established.

In in this paper we establish an explicit and sharp estimate of the spectral gap (Poincaré inequality) and the transportation inequality for Gibbs measures, under the Dobrushin uniqueness condition. Moreover, we give a generalization of the Liggett’s M−ɛ theorem for interacting particle systems.

We consider a system of random walks in a random environment interacting via exclusion. The model is reversible with respect to a family of disordered Bernoulli measures. Assuming some weak mixing conditions, it is shown that, under diffusive scaling, the system has a deterministic hydrodynamic limit which holds for almost every realization of the environment. The limit is a nonlinear diffusion equation with diffusion coefficient given by a variational formula. The model is nongradient and the method used is the “long jump” variation of the standard nongradient method, which is a type of renormalization. The proof is valid in all dimensions.

Free probability analogs of the basics of extreme-value theory are obtained, based on Ando’s spectral order. This includes classification of freely max-stable laws and their domains of attraction, using “free extremal convolutions” on the distributions. These laws coincide with the limit laws in the classical peaks-over-threshold approach. A free extremal projection-valued process over a measure-space is constructed, which is related to the free Poisson point process.