Our aim is to unify and extend the large deviation upper and lower bounds for the occupation times of a Markov process with L₂ semigroups under minimal conditions on the state space and the process trajectories; for example, no strong Markov property is needed. The methods used here apply in both continuous and discrete time. We present the proofs for continuous time only because of the inherent technical difficulties in that situation; the proofs can be adapted for discrete time in a straightforward manner.

This paper provides information about the asymptotic behavior of a one-dimensional Brownian polymer in random medium represented by a Gaussian field W on ℝ₊×ℝ which is white noise in time and function-valued in space. According to the behavior of the spatial covariance of W, we give a lower bound on the power growth (wandering exponent) of the polymer when the time parameter goes to infinity: the polymer is proved to be superdiffusive, with a wandering exponent exceeding any α<3/5.

We prove a quenched central limit theorem for random walks with bounded increments in a randomly evolving environment on ℤ^d. We assume that the transition probabilities of the walk depend not too strongly on the environment and that the evolution of the environment is Markovian with strong spatial and temporal mixing properties.

We consider a four-vertex model introduced by Bálint Tóth: a dependent bond percolation model on ℤ² in which every edge is present with probability 1/2 and each vertex has exactly two incident edges, perpendicular to each other. We prove that all components are finite cycles almost surely, but the expected diameter of the cycle containing the origin is infinite. Moreover, we derive the following critical exponents: the tail probability ℙ(diameter of the cycle of the origin >n)≈n^−γ and the expectation $\mathbb{E}$ (length of a typical cycle with diameter n)≈n^δ, with $\gamma=(5-\sqrt{17})/4=0.219\ldots$ and $\delta=(\sqrt{17}+1)/4=1.28\ldots$. The value of δ comes from a singular sixth order ODE, while the relation γ+δ=3/2 corresponds to the fact that the scaling limit of the natural height function in the model is the additive Brownian motion, whose level sets have Hausdorff dimension 3/2. We also include many open problems, for example, on the conformal invariance of certain linear entropy models.

We study gradient percolation for site percolation on the triangular lattice. This is a percolation model where the percolation probability depends linearly on the location of the site. We prove the results predicted by physicists for this model. More precisely, we describe the fluctuations of the interfaces around their (straight) scaling limits, and the expected and typical lengths of these interfaces. These results build on the recent results for critical percolation on this lattice by Smirnov, Lawler, Schramm and Werner, and on the scaling ideas developed by Kesten.

Let X=(X_t)_t≥0 be a stable Lévy process of index α∈(1, 2) with no negative jumps and let S_t=sup_0≤s≤t X_s denote its running supremum for t>0. We show that the density function f_t of S_t can be characterized as the unique solution to a weakly singular Volterra integral equation of the first kind or, equivalently, as the unique solution to a first-order Riemann–Liouville fractional differential equation satisfying a boundary condition at zero. This yields an explicit series representation for f_t. Recalling the familiar relation between S_t and the first entry time τ_x of X into [x, ∞), this further translates into an explicit series representation for the density function of τ_x.

Given any regularly varying dislocation measure, we identify a natural self-similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldous’s beta-splitting models and Ford’s alpha models for phylogenetic trees. This confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf.

We consider the following problem in one-dimensional diffusion-limited aggregation (DLA). At time t, we have an “aggregate” consisting of ℤ∩[0, R(t)] [with R(t) a positive integer]. We also have N(i, t) particles at i, i>R(t). All these particles perform independent continuous-time symmetric simple random walks until the first time t'>t at which some particle tries to jump from R(t)+1 to R(t). The aggregate is then increased to the integers in [0, R(t')]=[0, R(t)+1] [so that R(t')=R(t)+1] and all particles which were at R(t)+1 at time t' − are removed from the system. The problem is to determine how fast R(t) grows as a function of t if we start at time 0 with R(0)=0 and the N(i, 0) i.i.d. Poisson variables with mean μ>0. It is shown that if μ<1, then R(t) is of order $\sqrt{t}$, in a sense which is made precise. It is conjectured that R(t) will grow linearly in t if μ is large enough.

One of the most well-known classical results for site percolation on the square lattice is the equation p_c+p_c^*=1. In words, this equation means that for all values ≠ p_c of the parameter p, the following holds: either a.s. there is an infinite open cluster or a.s. there is an infinite closed “star” cluster. This result is closely related to the percolation transition being sharp: below p_c, the size of the open cluster of a given vertex is not only (a.s.) finite, but has a distribution with an exponential tail. The analog of this result has been proven by Higuchi in 1993 for two-dimensional Ising percolation (at fixed inverse temperature β<β_c) with external field h, the parameter of the model.

Using sharp-threshold results (approximate zero-one laws) and a modification of an RSW-like result by Bollobás and Riordan, we show that these results hold for a large class of percolation models where the vertex values can be “nicely” represented (in a sense which will be defined precisely) by i.i.d. random variables. We point out that the ordinary percolation model obviously belongs to this class and we also show that the Ising model mentioned above belongs to it.

Recently, in [Preprint (2006)], we extended the concept of intrinsic ultracontractivity to nonsymmetric semigroups. In this paper, we study the intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and measure-valued potentials in bounded domains. Our process Y is a diffusion process whose generator can be formally written as L+μ⋅∇−ν with Dirichlet boundary conditions, where L is a uniformly elliptic second-order differential operator and μ=(μ¹, …, μ^d) is such that each component μⁱ, i=1, …, d, is a signed measure belonging to the Kato class K_d,1 and ν is a (nonnegative) measure belonging to the Kato class K_d,2. We show that scale-invariant parabolic and elliptic Harnack inequalities are valid for Y.

In this paper, we prove the parabolic boundary Harnack principle and the intrinsic ultracontractivity for the killed diffusion Y^D with measure-valued drift and potential when D is one of the following types of bounded domains: twisted Hölder domains of order α∈(1/3, 1], uniformly Hölder domains of order α∈(0, 2) and domains which can be locally represented as the region above the graph of a function. This extends the results in [J. Funct. Anal. 100 (1991) 181–206] and [Probab. Theory Related Fields 91 (1992) 405–443]. As a consequence of the intrinsic ultracontractivity, we get that the supremum of the expected conditional lifetimes of Y^D is finite.

For a given one-dimensional random walk {S_n} with a subexponential step-size distribution, we present a unifying theory to study the sequences {x_n} for which $\mathsf{P}\{S_{n}>x\}\sim n\mathsf{P}\{S_{1}>x\}$ as n→∞ uniformly for x≥x_n. We also investigate the stronger “local” analogue, $\mathsf{P}\{S_{n}\in(x,x+T]\}\sim n\mathsf{P}\{S_{1}\in(x,x+T]\}$. Our theory is self-contained and fits well within classical results on domains of (partial) attraction and local limit theory.

When specialized to the most important subclasses of subexponential distributions that have been studied in the literature, we reproduce known theorems and we supplement them with new results.