We discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index α∈(1, 2). When the number n of vertices of the map tends to infinity, the asymptotic behavior of distances from a distinguished vertex is described by a random process called the continuous distance process, which can be constructed from a centered stable process with no negative jumps and index α. In particular, the profile of distances in the map, rescaled by the factor n^−1∕2α, converges to a random measure defined in terms of the distance process. With the same rescaling of distances, the vertex set viewed as a metric space converges in distribution as n→∞, at least along suitable subsequences, toward a limiting random compact metric space whose Hausdorff dimension is equal to 2α.

The vacant set of random interlacements on ℤ^d, d≥3, has nontrivial percolative properties. It is known from Sznitman [Ann. Math. 171 (2010) 2039–2087], Sidoravicius and Sznitman [Comm. Pure Appl. Math. 62 (2009) 831–858] that there is a nondegenerate critical value u_∗ such that the vacant set at level u percolates when u<u_∗ and does not percolate when u>u_∗. We derive here an asymptotic upper bound on u_∗, as d goes to infinity, which complements the lower bound from Sznitman [Probab. Theory Related Fields, to appear]. Our main result shows that u_∗ is equivalent to log d for large d and thus has the same principal asymptotic behavior as the critical parameter attached to random interlacements on 2d-regular trees, which has been explicitly computed in Teixeira [Electron. J. Probab. 14 (2009) 1604–1627].

We consider the family of two-sided Bernoulli initial conditions for TASEP which, as the left and right densities (ρ₋, ρ₊) are varied, give rise to shock waves and rarefaction fans—the two phenomena which are typical to TASEP. We provide a proof of Conjecture 7.1 of [Progr. Probab. 51 (2002) 185–204] which characterizes the order of and scaling functions for the fluctuations of the height function of two-sided TASEP in terms of the two densities ρ₋, ρ₊ and the speed y around which the height is observed.

In proving this theorem for TASEP, we also prove a fluctuation theorem for a class of corner growth processes with external sources, or equivalently for the last passage time in a directed last passage percolation model with two-sided boundary conditions: ρ₋ and 1−ρ₊. We provide a complete characterization of the order of and the scaling functions for the fluctuations of this model’s last passage time L(N, M) as a function of three parameters: the two boundary/source rates ρ₋ and 1−ρ₊, and the scaling ratio γ²=M∕N. The proof of this theorem draws on the results of [Comm. Math. Phys. 265 (2006) 1–44] and extensively on the work of [Ann. Probab. 33 (2005) 1643–1697] on finite rank perturbations of Wishart ensembles in random matrix theory.

In this paper, we study a model of a Brownian polymer in ℝ₊×ℝ^d, introduced by Rovira and Tindel [J. Funct. Anal. 222 (2005) 178–201]. Our investigation focuses mainly on the effect of strong spatial correlation in the environment in that model in terms of free energy, fluctuation exponent and volume exponent. In particular, we prove that under some assumptions, very strong disorder and superdiffusivity hold at all temperatures when d≥3 and provide a novel approach to Petermann’s superdiffusivity result in dimension one [Superdiffusivity of directed polymers in random environment (2000) Ph.D. thesis]. We also derive results for a Brownian model of pinning in a nonrandom potential with power-law decay at infinity.

In this paper, we study Bouchaud’s trap model on the discrete d-dimensional torus ${\mathbb{T}}^{d}_{n}=({\mathbb{Z}}/n{\mathbb{Z}})^{d}$. In this process, a particle performs a symmetric simple random walk, which waits at the site $x\in {\mathbb{T}}^{d}_{n}$ an exponential time with mean ξ_x, where $\{\xi_{x},x\in {\mathbb{T}}^{d}_{n}\}$ is a realization of an i.i.d. sequence of positive random variables with an α-stable law. Intuitively speaking, the value of ξ_x gives the depth of the trap at x. In dimension d=1, we prove that a system of independent particles with the dynamics described above has a hydrodynamic limit, which is given by the degenerate diffusion equation introduced in [Ann. Probab. 30 (2002) 579–604]. In dimensions d>1, we prove that the evolution of a single particle is metastable in the sense of Beltrán and Landim [Tunneling and Metastability of continuous time Markov chains (2009) Preprint]. Moreover, we prove that in the ergodic scaling, the limiting process is given by the K-process, introduced by Fontes and Mathieu in [Ann. Probab. 36 (2008) 1322–1358].

We consider gradient fields (ϕ_x : x∈ℤ^d) whose law takes the Gibbs–Boltzmann form Z⁻¹exp{−∑_〈x, y〉V(ϕ_y−ϕ_x)}, where the sum runs over nearest neighbors. We assume that the potential V admits the representation

V(η):=−log∫ϱ(d κ)exp[−½κη²],

where ϱ is a positive measure with compact support in (0, ∞). Hence, the potential V is symmetric, but nonconvex in general. While for strictly convex V’s, the translation-invariant, ergodic gradient Gibbs measures are completely characterized by their tilt, a nonconvex potential as above may lead to several ergodic gradient Gibbs measures with zero tilt. Still, every ergodic, zero-tilt gradient Gibbs measure for the potential V above scales to a Gaussian free field.

In this paper we introduce a critical curve separating the asymptotic behavior of the moments of the symbiotic branching model, introduced by Etheridge and Fleischmann [Stochastic Process. Appl. 114 (2004) 127–160] into two regimes. Using arguments based on two different dualities and a classical result of Spitzer [Trans. Amer. Math. Soc. 87 (1958) 187–197] on the exit-time of a planar Brownian motion from a wedge, we prove that the parameter governing the model provides regimes of bounded and exponentially growing moments separated by subexponential growth. The moments turn out to be closely linked to the limiting distribution as time tends to infinity. The limiting distribution can be derived by a self-duality argument extending a result of Dawson and Perkins [Ann. Probab. 26 (1998) 1088–1138] for the mutually catalytic branching model.

As an application, we show how a bound on the 35th moment improves the result of Etheridge and Fleischmann [Stochastic Process. Appl. 114 (2004) 127–160] on the speed of the propagation of the interface of the symbiotic branching model.

We establish a version of the Feynman–Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. We use the techniques of Malliavin calculus to prove that the process defined by the Feynman–Kac formula is a weak solution of the stochastic heat equation. From the Feynman–Kac formula, we establish the smoothness of the density of the solution and the Hölder regularity in the space and time variables. We also derive a Feynman–Kac formula for the stochastic heat equation in the Skorokhod sense and we obtain the Wiener chaos expansion of the solution.

Consider a critical nearest-neighbor branching random walk on the d-dimensional integer lattice initiated by a single particle at the origin. Let G_n be the event that the branching random walk survives to generation n. We obtain the following limit theorems, conditional on the event G_n, for a variety of occupation statistics: (1) Let V_n be the maximal number of particles at a single site at time n. If the offspring distribution has finite αth moment for some integer α≥2, then, in dimensions 3 and higher, V_n=O_p(n^1∕α). If the offspring distribution has an exponentially decaying tail, then V_n=O_p(log n) in dimensions 3 and higher and V_n=O_p((log n)²) in dimension 2. Furthermore, if the offspring distribution is nondegenerate, then P(V_n≥δlog n|G_n)→1 for some δ>0. (2) Let M_n(j) be the number of multiplicity-j sites in the nth generation, that is, sites occupied by exactly j particles. In dimensions 3 and higher, the random variables M_n(j)∕n converge jointly to multiples of an exponential random variable. (3) In dimension 2, the number of particles at a “typical” site (i.e., at the location of a randomly chosen particle of the nth generation) is of order O_p(log n) and the number of occupied sites is O_p(n∕log n). We also show that, in dimension 2, there is particle clustering around a typical site.

We consider a class of stochastic reaction-diffusion equations also having a stochastic perturbation on the boundary and we show that when the diffusion rate is much larger than the rate of reaction, it is possible to replace the SPDE by a suitable one-dimensional stochastic differential equation. This replacement is possible under the assumption of spectral gap for the diffusion and is a result of averaging in the fast spatial transport. We also study the fluctuations around the averaged motion.