We characterize the law of the partition function of a Brownian directed polymer model in terms of a diffusion process associated with the quantum Toda lattice. The proof is via a multidimensional generalization of a theorem of Matsumoto and Yor concerning exponential functionals of Brownian motion. It is based on a mapping which can be regarded as a geometric variant of the RSK correspondence.

Consider a random walk in an i.i.d. uniformly elliptic environment in dimensions larger than one. In 2002, Sznitman introduced for each γ ∈ (0, 1) the ballisticity condition (T)_γ and the condition (T') defined as the fulfillment of (T)_γ for each γ ∈ (0, 1). Sznitman proved that (T') implies a ballistic law of large numbers. Furthermore, he showed that for all γ ∈ (0.5, 1), (T)_γ is equivalent to (T'). Recently, Berger has proved that in dimensions larger than three, for each γ ∈ (0, 1), condition (T)_γ implies a ballistic law of large numbers. On the other hand, Drewitz and Ramírez have shown that in dimensions d ≥ 2 there is a constant γ_d ∈ (0.366, 0.388) such that for each γ ∈ (γ_d, 1), condition (T)_γ is equivalent to (T'). Here, for dimensions larger than three, we extend the previous range of equivalence to all γ ∈ (0, 1). For the proof, the so-called effective criterion of Sznitman is established employing a sharp estimate for the probability of atypical quenched exit distributions of the walk leaving certain boxes. In this context, we also obtain an affirmative answer to a conjecture raised by Sznitman in 2004 concerning these probabilities. A key ingredient for our estimates is the multiscale method developed recently by Berger.

We show that the measure on markings of Z_n^d, d ≥ 3, with elements of {0, 1} given by i.i.d. fair coin flips on the range $\mathcal{R}$ of a random walk X run until time T and 0 otherwise becomes indistinguishable from the uniform measure on such markings at the threshold T = ½T_cov(Z_n^d). As a consequence of our methods, we show that the total variation mixing time of the random walk on the lamplighter graph Z₂ ≀ Z_n^d, d ≥ 3, has a cutoff with threshold ½T_cov(Z_n^d). We give a general criterion under which both of these results hold; other examples for which this applies include bounded degree expander families, the intersection of an infinite supercritical percolation cluster with an increasing family of balls, the hypercube and the Caley graph of the symmetric group generated by transpositions. The proof also yields precise asymptotics for the decay of correlation in the uncovered set.

We show that, under mild assumptions on the limiting curve, a sequence of simple chordal planar curves converges uniformly whenever certain Loewner driving functions converge. We extend this result to random curves. The random version applies in particular to random lattice paths that have chordal SLE_κ as a scaling limit, with κ < 8 (nonspace-filling).

Existing SLE_κ convergence proofs often begin by showing that the Loewner driving functions of these paths (viewed from ∞) converge to Brownian motion. Unfortunately, this is not sufficient, and additional arguments are required to complete the proofs. We show that driving function convergence is sufficient if it can be established for both parametrization directions and a generic observation point.

The arctic circle theorem of Jockusch, Propp, and Shor asserts that uniformly random domino tilings of an Aztec diamond of high order are frozen with asymptotically high probability outside the “arctic circle” inscribed within the diamond. A similar arctic circle phenomenon has been observed in the limiting behavior of random square Young tableaux. In this paper, we show that random domino tilings of the Aztec diamond are asymptotically related to random square Young tableaux in a more refined sense that looks also at the behavior inside the arctic circle. This is done by giving a new derivation of the limiting shape of the height function of a random domino tiling of the Aztec diamond that uses the large-deviation techniques developed for the square Young tableaux problem in a previous paper by Pittel and the author. The solution of the variational problem that arises for domino tilings is almost identical to the solution for the case of square Young tableaux by Pittel and the author. The analytic techniques used to solve the variational problem provide a systematic, guess-free approach for solving problems of this type which have appeared in a number of related combinatorial probability models.

Let X be an irreducible shift of finite type (SFT) of positive entropy, and let B_n(X) be its set of words of length n. Define a random subset ω of B_n(X) by independently choosing each word from B_n(X) with some probability α. Let X_ω be the (random) SFT built from the set ω. For each 0 ≤ α ≤ 1 and n tending to infinity, we compute the limit of the likelihood that X_ω is empty, as well as the limiting distribution of entropy for X_ω. For α near 1 and n tending to infinity, we show that the likelihood that X_ω contains a unique irreducible component of positive entropy converges exponentially to 1. These results are obtained by studying certain sequences of random directed graphs. This version of “random SFT” differs significantly from a previous notion by the same name, which has appeared in the context of random dynamical systems and bundled dynamical systems.

We study the asymptotic behavior of a self-interacting one-dimensional Brownian polymer first introduced by Durrett and Rogers [Probab. Theory Related Fields 92 (1992) 337–349]. The polymer describes a stochastic process with a drift which is a certain average of its local time.

We show that a smeared out version of the local time function as viewed from the actual position of the process is a Markov process in a suitably chosen function space, and that this process has a Gaussian stationary measure. As a first consequence, this enables us to partially prove a conjecture about the law of large numbers for the end-to-end displacement of the polymer formulated in Durrett and Rogers [Probab. Theory Related Fields 92 (1992) 337–349].

Next we give upper and lower bounds for the variance of the process under the stationary measure, in terms of the qualitative infrared behavior of the interaction function. In particular, we show that in the locally self-repelling case (when the process is essentially pushed by the negative gradient of its own local time) the process is super-diffusive.

We consider a model of a polymer in ℤ^d+1, constrained to join 0 and a hyperplane at distance N. The polymer is subject to a quenched nonnegative random environment. Alternatively, the model describes crossing random walks in a random potential (see Zerner [Ann Appl. Probab. 8 (1998) 246–280] or Chapter 5 of Sznitman [Brownian Motion, Obstacles and Random Media (1998) Springer] for the original Brownian motion formulation). It was recently shown [Ann. Probab. 36 (2008) 1528–1583; Probab. Theory Related Fields 143 (2009) 615–642] that, in such a setting, the quenched and annealed free energies coincide in the limit N → ∞, when d ≥ 3 and the temperature is sufficiently high. We first strengthen this result by proving that, under somewhat weaker assumptions on the distribution of disorder which, in particular, enable a small probability of traps, the ratio of quenched and annealed partition functions actually converges. We then conclude that, in this case, the polymer obeys a diffusive scaling, with the same diffusivity constant as the annealed model.

Let E be the set of edges of the d-dimensional cubic lattice ℤ^d, with d ≥ 2, and let t(e), e ∈ E, be nonnegative values. The passage time from a vertex v to a vertex w is defined as inf_{π : v→w} ∑_e∈π t(e), where the infimum is over all paths π from v to w, and the sum is over all edges e of π.

Benjamini, Kalai and Schramm [2] proved that if the t(e)’s are i.i.d. two-valued positive random variables, the variance of the passage time from the vertex 0 to a vertex v is sublinear in the distance from 0 to v. This result was extended to a large class of independent, continuously distributed t-variables by Benaïm and Rossignol [1].

We extend the result by Benjamini, Kalai and Schramm in a very different direction, namely to a large class of models where the t(e)’s are dependent. This class includes, among other interesting cases, a model studied by Higuchi and Zhang [9], where the passage time corresponds with the minimal number of sign changes in a subcritical “Ising landscape.”

Let X be a spectrally negative self-similar Markov process with 0 as an absorbing state. In this paper, we show that the distribution of the absorption time is absolutely continuous with an infinitely continuously differentiable density. We provide a power series and a contour integral representation of this density. Then, by means of probabilistic arguments, we deduce some interesting analytical properties satisfied by these functions, which include, for instance, several types of hypergeometric functions. We also give several characterizations of the Kesten’s constant appearing in the study of the asymptotic tail distribution of the absorbtion time. We end the paper by detailing some known and new examples. In particular, we offer an alternative proof of the recent result obtained by Bernyk, Dalang and Peskir [Ann. Probab. 36 (2008) 1777–1789] regarding the law of the maximum of spectrally positive Lévy stable processes.

In this article we prove a maximal L^p-regularity result for stochastic convolutions, which extends Krylov’s basic mixed L^p(L^q)-inequality for the Laplace operator on ℝ^d to large classes of elliptic operators, both on ℝ^d and on bounded domains in ℝ^d with various boundary conditions. Our method of proof is based on McIntosh’s H^∞-functional calculus, R-boundedness techniques and sharp L^p(L^q)-square function estimates for stochastic integrals in L^q-spaces. Under an additional invertibility assumption on A, a maximal space–time L^p-regularity result is obtained as well.

We first prove some general results on pathwise uniqueness, comparison property and existence of nonnegative strong solutions of stochastic equations driven by white noises and Poisson random measures. The results are then used to prove the strong existence of two classes of stochastic flows associated with coalescents with multiple collisions, that is, generalized Fleming–Viot flows and flows of continuous-state branching processes with immigration. One of them unifies the different treatments of three kinds of flows in Bertoin and Le Gall [Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 307–333]. Two scaling limit theorems for the generalized Fleming–Viot flows are proved, which lead to sub-critical branching immigration superprocesses. From those theorems we derive easily a generalization of the limit theorem for finite point motions of the flows in Bertoin and Le Gall [Illinois J. Math. 50 (2006) 147–181].

Let E be a locally compact separable metric space and m be a positive Radon measure on it. Given a nonnegative function k defined on E × E off the diagonal whose anti-symmetric part is assumed to be less singular than the symmetric part, we construct an associated regular lower bounded semi-Dirichlet form η on L²(E; m) producing a Hunt process X⁰ on E whose jump behaviours are governed by k. For an arbitrary open subset D ⊂ E, we also construct a Hunt process X^D,0 on D in an analogous manner. When D is relatively compact, we show that X^D,0 is censored in the sense that it admits no killing inside D and killed only when the path approaches to the boundary. When E is a d-dimensional Euclidean space and m is the Lebesgue measure, a typical example of X⁰ is the stable-like process that will be also identified with the solution of a martingale problem up to an η-polar set of starting points. Approachability to the boundary ∂ D in finite time of its censored process X^D,0 on a bounded open subset D will be examined in terms of the polarity of ∂ D for the symmetric stable processes with indices that bound the variable exponent α(x).