We study uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free (FSF )or wired (WSF ) boundary conditions. Pemantle proved that the free and wired spanning forests coincide in $\mathbb{Z}^d$ and that they give a single tree iff $d </4$.

In the present work, we extend Pemantle’s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks, potential theory, invariant percolation and amenability. The uniform spanning forest model is related to random cluster models in statistical physics, but, because of the preceding connections, its analysis can be carried further. Among our results are the following:

• The FSF and WSF in a graph G coincide iff all harmonic Dirichlet functions on G are constant.

• The tail $\sigma$-fields of the WSF and the FSF are trivial on any graph.

• On any Cayley graph that is not a finite extension of f $\mathbb{Z}$ all component trees of the WSF have one end; this is new in $\mathbb{Z}^d$ for $d \ge 5$.

• On any tree, as well as on any graph with spectral radius less than 1, a.s. all components of the WSF are recurrent.

• The basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space $\mathbb{H}^d$ is analyzed.

• A Cayley graph is amenable iff for all $\epsilon > 0$, the union of the WSF and Bernoulli percolation with parameter $\epsilon$ is connected.

• Harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees.

We also present numerous open problems and conjectures.

Vertex-reinforced random walk (VRRW), defined by Pemantle, is a random process in a continuously changing environment which is more likely to visit states it has visited before. We consider VRRW on arbitrary graphs and show that on almost all of them, VRRW visits only finitely many vertices with a positive probability. We conjecture that on all graphs of bounded degree, this happens with probability 1, and provide a proof only for trees of this type.

We distinguish between several different patterns of localization and explicitly describe the long-run structure of VRRW, which depends on whether a graph contains triangles or not.

While the results of this paper generalize those obtained by Pemantle and Volkov for Z¹, ideas of proofs are different and typically based on a large deviation principle rather than a martingale approach.

Consider a translation-invariant bond percolation model on the integer lattice which has exponential decay of connectivities, that is, the probability of a connection $0 \leftrightarrow x$ by a path of open bonds decreases like $\exp\{-m(\theta)|x|\}$ for some positive constant $m(\theta)$ which may depend on the direction $\theta = x/|x|$. In two and three dimensions, it is shown that if the model has an appropriate mixing property and satisfies a special case of the FKG property, then there is at most a power-law correction to the exponential decay—there exist $A$ and $C$ such that $\exp\{-m(\theta)|x|\} \ge P(0 \leftrightarrow x) \ge A|x|^{-C} \exp\{-m(\theta)|x|\}$ for all nonzero $x$ . In four or more dimensions, a similar bound holds with $|x|^{-C}$ replaced by $\exp\{-C(\log |x|)^2\}$. In particular the power-law lower bound holds for the Fortuin-Kasteleyn random cluster model in two dimensions whenever the connectivity decays exponentially, since the mixing property is known to hold in that case. Consequently a similar bound holds for correlations in the Potts model at supercritical temperatures.

It is well-known in percolation theory (and intuitively plausible) that two events of the form “there is an open path from s to a ”are positively correlated. We prove the (not intuitively obvious) fact that this is still true if we condition on an event of the form “there is no open path from s to t.”

We prove uniqueness of the infinite entangled component for bond percolation on the three-dimensional cubic lattice above the entanglement critical probability. This improves earlier results by Grimmett and Holroyd.

We consider an interacting random walk on $\mathbb{Z}^d$ where particles interact only at times when a particle jumps to a site at which there are at least $n - 1$ other particles present. If there are $i \ge n - 1$ particles present, then the particle coalesces (is removed from the system) with probability $c_i$ and annihilates (is removed along with another particle) with probability $a_i$. We call this process the $n$-threshold randomly coalescing and annihilating random walk. We show that, for $n \ge 3$, if both $a_i$ and $a_i + c_i$ are increasing in $i$ and if the dimension $d$ is at least $2n + 4$, then $$P\{\text{the origin is occupied at time $t$}\}\sim C(d, n) t^{−\frac{1}{n-1}},\\ E\{\text{number of particles at the origin at time $t$}\} \sim C(d, n) t^{−\frac{1}{n-1}}.$$ The constants $C(d, n)$ are explicitly identified. The proof is an extension of a result obtained by Kesten and van den Berg for the 2-threshold coalescing random walk and is based on an approximation for $dE(t)/dt$.

We study the evolution of a small perturbation of the equilibrium of a totally asymmetric one-dimensional interacting system. The model we take as an example is Hammersley's process as seen from a tagged particle, which can be viewed as a process of interacting positive-valued stick heights on the sites of $\mathbf{Z}$. It is known that under Euler scaling (space and time scale $n$) the empirical stick profile obeys the Burgers equation. We refine this result in two ways. If the process starts close enough to equilibrium, then over times $n^\nu$ for $1 \le \nu < 3$, and up to errors that vanish in hydrodynamic scale, the dynamics merely translates the initial stick configuration. In particular, on the hydrodynamic time scale, diffusive fluctuations are translated rigidly. A time evolution for the perturbation is visible under a particular family of scalings:over times $n_{\nu}, 1 < \nu < 3/2$, a perturbation of order $n^{1-\nu}$ from equilibrium follows the inviscid Burgers equation. The results for the stick model are derived from asymptotic results for tagged particles in Hammersley's process.

Let $\{X_v, v \in \mathbb{Z}^d\}$ be i.i.d. random variables, and $S(\xi) = \sum_{v \in \xi} X_v$ be the weight of a lattice animal $\xi$. Let $N_n = \max\{S(\xi) : |\xi| = n$ and $\xi$ contains the origin} and $G_n = \max\{S(\xi) : \xi \subseteq [-n,n]^d\}$. We show that, regardless of the negative tail of the distribution of $X_v$ , if $\mathbf{E}( X_v^+)^d (\log^+ X_v^+))^{d+a} < + \infty$ for some $a>0$, then first, $\lim_n n^{-1} N_n = N$ exists, is finite and constant a.e.; and, second, there is a transition in the asymptotic behavior of $G_n$ depending on the sign of $N$: if $N > 0$ then $G_n \approx n^d$, and if $N < 0$ then $G_n \le cn$, for some $c > 0$. The exact behavior of $G_n$ in this last case depends on the positive tail of the distribution of $X_v$; we show that if it is nontrivial and has exponential moments, then $G_n \approx \log n$, with a transition from $G_n \approx n^d$ occurring in general not as predicted by large deviations estimates. Finally, if $x^d(1 - F(x)) \to \infty \text{ as }x \to \infty$, then no transition takes place.

We continue our study of large deviations of the empirical measures of a massless Gaussian field on $\mathbb Z^d$, whose covariance is given by the Green function of a long-range random walk. In this paper we extend techniques and results of Bolthausen and Deuschel to the nonlocal case of a random walk in the domain of attraction of the symmetric $\alpha$-stable law, with $\alpha \in (0, 2 \wedge d)$. In particular, we show that critical large deviations occur at the capacity scale $N^{d-\alpha}$, with a rate function given by the Dirichlet form of the embedded $\alpha$-stable process. We also prove that if we impose zero boundary conditions, the rate function is given by the Dirichlet form of the killed $\alpha$- stable process.

We consider a random walk on $S^{n-1}$ , the standard sphere of dimension $n -1$, generated by random rotations on randomly selected coordinate planes $i,j$ with $1 \le i < j \le n$. This dynamic was used by Marc Kac as a model for the spatially homogeneous Boltzmann equation. We prove that the spectral gap on $S^{n-1}$ is $n^{-1}$ up to a constant independent of $n$.

We present a Kolmogorov’s test for the Brownian snake. This result was conjectured by Le Gall in 1998. It has to be compared with Kolmogorov’s test for super Brownian motion by Dhersin and Le Gall.

We construct a continuous superprocess X on M (R ^d) which is the unique weak Feller extension of the empirical process of consistent k-point motions generated by a family of differential operators. The process X differs from known Dawson–Watanabe type, Fleming–Viot type and Ornstein–Uhlenbeck type superprocesses. This new type of superprocess provides a connection between stochastic flows and measure-valued processes, and determines a stochastic coalescence which is similar to those of Smoluchowski. Moreover, the support of X describes how an initial measure on R ^d is transported under the flow. As an example, the process realizes a viewpoint of Darling about the isotropic stochastic flows under certain conditions.

Aldous and Pitman have studied the asymptotic behavior of the additive coalescent processes using a nested family random forests derived by logging certain inhomogeneous continuum random trees. Here we propose a different approach based on partitions of the unit interval induced by certain bridges with exchangeable increments. The analysis is made simple by an interpretation in terms of an aggregating server system.

The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge $(B^{\mathrm{br}}_t, 0 \le t \le 1)$ is described. The height $M^{\mathrm{br} +}_j$of the $j$th highest maximum over a positive excursion of the bridge has the same distribution as $M^{\mathrm{br} +}_1 /j$, where the distribution of $M^{\mathrm{br} +}_1 = \sup_{0 \le t \le 1} B^{\mathrm{br}}_t$ is given by Lévy’s formula $P(M^{\mathrm{br} +}_1 > x) = e^{−2x^{2}}$. The probability density of the height $M^{\mathrm{br}}_j$ of the $j$th highest maximum of excursions of the reflecting Brownian bridge $(|B^{\mathrm{br}}_t|, 0 \le t \le 1)$ is given by a modification of the known $\theta$-function series for the density of $M^{\mathrm{br}}_1 = \sup_{0 \le t \le 1} |B^{\mathrm{br}}_t|$. These results are obtained from a more general description of the distribution of ranked values of a homogeneous functional of excursions of the standardized bridge of a self-similar recurrent Markov process.

The distributions of stochastic integrals are approximated by the distributions of stochastic integrals of piece-wise constant processes. The rate of approximation in some negative Sobolev spaces is estimated. Generalizations are given for problems arising in control theory.

We give a reformulation of majorizing measures that does not involve measures, but rather special sequences of partitions. This formulation is more convenient to perform chaining with different distances.

In the frame of Borel right Markov processes, we investigate, following an analytical point of view, the Revuz correspondence between classes of potential kernels and their associated measures, improving upon the results of Revuz, Azéma, Getoor and Sharpe, Fitzsimmons, Fitzsimmons and Getoor and Dellacherie, Maisonneuve and Meyer. In the probabilistic approach of the problem, the kernels that occur are the potential operators of different types of homogeneous random measures. We completely characterize the hypothesis (B) of Hunt in terms of Revuz measures.

We consider a Markov chain on a locally compact metric space with an absorbing set. Necessary and sufficient conditions are provided for the existence of a quasistationary probability distribution.

This paper considers how to measure the magnitude of the sum of independent random variables in several ways. We give a formula for the tail distribution for sequences that satisfy the so called Lévy property. We then give a connection between the tail distribution and the pth moment, and between the pth moment and the rearrangement invariant norms.

The structure of extreme value distributions in in finite-dimensional space is well known. We characterize the domain of attraction of such extreme-value distributions in the framework of Giné Hahn and Vatan. We intend to use the result for statistical applications.

We prove that, with probability 1, eventually there are no more than three favorite (i.e., most visited) sites of simple symmetric random walks. This partially answers a relatively longstanding question of Erdös and Révész.

We analyze and identify stationary fields with linear regressions and quadratic conditional variances. We give sufficient conditions to determine one-dimensional distributions uniquely as normal and as certain compactly supported distributions. Our technique relies on orthogonal polynomials, which under our assumptions turn out to be a version of the so-called continuous q-Hermite polynomials.

Let $X, X_i, i \in \mathbf{N}$, be independent identically distributed random variables and let $h(x,y) = h(y,x)$ be a measurable function of two variables. It is shown that the bounded law of the iterated logarithm, $\limsup _n \log n(\log n)^{-1}|\sum_{1\le i<j\le n} h(X_i, X_j)| < \infty$ a.s., holds if and only if the following three conditions are satisfied: $h$ is canonical for the law of $X$ [i.e., $Eh(X,y) = 0$ for almost all $y$] and there exists $C < \infty$ such that both $E(h^2(X_1,X_2) \wedge u) \le C \log \log u$ for all large $u$ and $\sup\{Eh(X_1,X_2) \times f(X_1)g(X_2) : ||f(X)||_2 \le 1, ||g(X)||_2 \le 1; ||f||_\infty < \infty, ||g||_\infty < \infty\} \le C$.

Let $F,G$ be arbitrary distribution functions on the real line and let $\widehat{F},\widehat{G}$ denote their respective bilateral Laplace transforms. Let $\kappa > 0$ and let $h : \mathbb{R}^+ \to \mathbb{R}^+$ be continuous, non-decreasing, and such that $h(u) \ge Au^4$ for some $A > 0$ and all $u \ge 0$. Under the assumptions that $$\sup_{0 \le u \le \kappa}|\widehat{F}(u) - \widehat{G} (u) | \le \varepsilon, \qquad \widehat{F}(u) + \widehat{G} (u) \le h(u) \qquad (-L \le u \le L),$$ we establish the bound $$\sup_{u \in \mathbb{R}}|\widehat{F}(u) - \widehat{G} (u) | \le CQ_G(l)$$ where $C$ is a constant depending at most on $\kappa$ and $A$, $Q_G$ is the concentration function of $G$, and $l := (\log L) /L + (\log W) /W$, with $W$ any solution to $h(W) = 1/\epsilon$. Improving and generalizing an estimate of Alladi, this result provides a Laplace transform analogue to the Berry-Esseen inequality, related to Fourier transforms. The dependence in $\epsilon$ is optimal up to the logarithmic factor log $W$. A number-theoretic application, developed in detail elsewhere, is described. It concerns so-called lexicographic integers, whose characterizing property is that their divisors are ranked according to size and valuation of the largest prime factor. The above inequality furnishes, among other informations, an effective Erdös-Kac theorem for lexicographical integers.