We consider a cluster growth model on ${\mathbb{Z}}^{d}$, called internal diffusion limited aggregation (internal DLA). In this model, random walks start at the origin, one at a time, and stop moving when reaching a site not occupied by previous walks. It is known that the asymptotic shape of the cluster is spherical. When dimension is 2 or more, we prove that fluctuations with respect to a sphere are at most a power of the logarithm of its radius in dimension $d\ge2$. In so doing, we introduce a closely related cluster growth model, that we call the flashing process, whose fluctuations are controlled easily and accurately. This process is coupled to internal DLA to yield the desired bound. Part of our proof adapts the approach of Lawler, Bramson and Griffeath, on another space scale, and uses a sharp estimate (written by Blachère in our Appendix) on the expected time spent by a random walk inside an annulus.

We consider internal diffusion limited aggregation in dimension larger than or equal to two. This is a random cluster growth model, where random walks start at the origin of the $d$-dimensional lattice, one at a time, and stop moving when reaching a site that is not occupied by previous walks. It is known that the asymptotic shape of the cluster is a sphere. When the dimension is two or more, we have shown in a previous paper that the inner (resp., outer) fluctuations of its radius is at most of order $\log(\mathrm{radius})$ [resp., $\log^{2}(\mathrm{radius})$]. Using the same approach, we improve the upper bound on the inner fluctuation to $\sqrt{\log(\mathrm{radius})}$ when $d$ is larger than or equal to three. The inner fluctuation is then used to obtain a similar upper bound on the outer fluctuation.

We show by example that there is a Cayley graph, having two invariant random subgraphs $X$ and $Y$, such that there exists a monotone coupling between them in the sense that $X\subset Y$, although no such coupling can be invariant. Here, “invariant” means that the distribution is invariant under group multiplications.

We show that the law of the overall supremum $\overline{X}_{t}=\sup_{s\let}X_{s}$ of a Lévy process $X$, before the deterministic time $t$ is equivalent to the average occupation measure $\mu_{t}^{+}(dx)=\int_{0}^{t}\mathbb{P} (X_{s}\in dx)\,ds$, whenever 0 is regular for both open halflines $(-\infty,0)$ and $(0,\infty)$. In this case, $\mathbb{P} (\overline{X}_{t}\in dx)$ is absolutely continuous for some (and hence for all) $t>0$ if and only if the resolvent measure of $X$ is absolutely continuous. We also study the cases where 0 is not regular for both halflines. Then we give absolute continuity criterions for the laws of $(g_{t},\overline{X}_{t})$ and $(g_{t},\overline{X}_{t},X_{t})$, where $g_{t}$ is the time at which the supremum occurs before $t$. The proofs of these results use an expression of the joint law $\mathbb{P} (g_{t}\in ds,X_{t}\in dx,\overline{X}_{t}\in dy)$ in terms of the entrance law of the excursion measure of the reflected process at the supremum and that of the reflected process at the infimum. As an application, this law is made (partly) explicit in some particular instances.

Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study the following “anisotropic” bootstrap percolation model: the neighborhood of a point $(m,n)$ is the set

\[\{(m+2,n),(m+1,n),(m,n+1),(m-1,n),(m-2,n),(m,n-1)\}.\]

At time 0, sites are occupied with probability $p$. At each time step, sites that are occupied remain occupied, while sites that are not occupied become occupied if and only if three of more sites in their neighborhood are occupied. We prove that it exhibits a sharp metastability threshold. This is the first mathematical proof of a sharp threshold for an anisotropic bootstrap percolation model.

We consider the weakly asymmetric limit of simple exclusion process with drift to the left, starting from step Bernoulli initial data with $\rho_{-}<\rho_{+}$ so that macroscopically one has a rarefaction fan. We study the fluctuations of the process observed along slopes in the fan, which are given by the Hopf–Cole solution of the Kardar–Parisi–Zhang (KPZ) equation, with appropriate initial data. For slopes strictly inside the fan, the initial data is a Dirac delta function and the one point distribution functions have been computed in [Comm. Pure Appl. Math. 64 (2011) 466–537] and [Nuclear Phys. B 834 (2010) 523–542]. At the edge of the rarefaction fan, the initial data is one-sided Brownian. We obtain a new family of crossover distributions giving the exact one-point distributions of this process, which converge, as $T\nearrow\infty$ to those of the Airy $\mathcal{A}_{2\to \mathrm{BM}}$ process. As an application, we prove moment and large deviation estimates for the equilibrium Hopf–Cole solution of KPZ. These bounds rely on the apparently new observation that the FKG inequality holds for the stochastic heat equation. Finally, via a Feynman–Kac path integral, the KPZ equation also governs the free energy of the continuum directed polymer, and thus our formula may also be interpreted in those terms.

In many spin glass models, due to the symmetry among sites, any limiting joint distribution of spins under the annealed Gibbs measure admits the Aldous–Hoover representation encoded by a function $\sigma : [0,1]^{4}\to\{-1,+1\}$, and one can think of this function as a generic functional order parameter of the model. In a class of diluted models, and in the Sherrington–Kirkpatrick model, we introduce novel perturbations of the Hamiltonian that yield certain invariance and self-consistency equations for this generic functional order parameter and we use these invariance properties to obtain representations for the free energy in terms of $\sigma$. In the setting of the Sherrington–Kirkpatrick model, the self-consistency equations imply that the joint distribution of spins is determined by the joint distributions of the overlaps, and we give an explicit formula for $\sigma$ under the Parisi ultrametricity hypothesis. In addition, we discuss some connections with the Ghirlanda–Guerra identities and stochastic stability and describe the expected Parisi ansatz in the diluted models in terms of $\sigma$.

We consider the minimum of a super-critical branching random walk. Addario-Berry and Reed [Ann. Probab. 37 (2009) 1044–1079] proved the tightness of the minimum centered around its mean value. We show that a convergence in law holds, giving the analog of a well-known result of Bramson [Mem. Amer. Math. Soc. 44 (1983) iv+190] in the case of the branching Brownian motion.

We investigate the convergence of McKean–Vlasov diffusions in a nonconvex landscape. These processes are linked to nonlinear partial differential equations. According to our previous results, there are at least three stationary measures under simple assumptions. Hence, the convergence problem is not classical like in the convex case. By using the method in Benedetto et al. [J. Statist. Phys. 91 (1998) 1261–1271] about the monotonicity of the free-energy, and combining this with a complete description of the set of the stationary measures, we prove the global convergence of the self-stabilizing processes.

Laws of large numbers, starting from certain nonequilibrium measures, have been shown for the integrated current across a bond, and a tagged particle in one-dimensional symmetric nearest-neighbor simple exclusion [Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 567–577]. In this article, we prove corresponding large deviation principles and evaluate the rate functions, showing different growth behaviors near and far from their zeroes which connect with results in [J. Stat. Phys. 136 (2009) 1–15].

In this paper we define and prove of the existence of the multi-point Green’s function for $\mbox{SLE}$—a normalized limit of the probability that an $\mbox{SLE}_{\kappa}$ curve passes near to a pair of marked points in the interior of a domain. When $\kappa<8$ this probability is nontrivial, and an expression can be written in terms two-sided radial $\mbox{SLE}$. One of the main components to our proof is a refinement of a bound first provided by Beffara [Ann. Probab. 36 (2008) 1421–1452]. This work contains a proof of this bound independent from the original.

Developing the theory of two-sided radial and chordal $\mathit{SLE}$, we prove that the natural parametrization on $\mathit{SLE}_{\kappa}$ curves is well defined for all $\kappa<8$. Our proof uses a two-interior-point local martingale.

Guided by the relationship between the breadth-first walk of a rooted tree and its sequence of generation sizes, we are able to include immigration in the Lamperti representation of continuous-state branching processes. We provide a representation of continuous-state branching processes with immigration by solving a random ordinary differential equation driven by a pair of independent Lévy processes. Stability of the solutions is studied and gives, in particular, limit theorems (of a type previously studied by Grimvall, Kawazu and Watanabe and by Li) and a simulation scheme for continuous-state branching processes with immigration. We further apply our stability analysis to extend Pitman’s limit theorem concerning Galton–Watson processes conditioned on total population size to more general offspring laws.

In this paper, we consider two skew Brownian motions, driven by the same Brownian motion, with different starting points and different skewness coefficients. We show that we can describe the evolution of the distance between the two processes with a stochastic differential equation. This S.D.E. possesses a jump component driven by the excursion process of one of the two skew Brownian motions. Using this representation, we show that the local time of two skew Brownian motions at their first hitting time is distributed as a simple function of a Beta random variable. This extends a result by Burdzy and Chen [Ann. Probab. 29 (2001) 1693–1715], where the law of coalescence of two skew Brownian motions with the same skewness coefficient is computed.

In this article, we derive a Stratonovich and Skorohod-type change of variables formula for a multidimensional Gaussian process with low Hölder regularity $\gamma$ (typically $\gamma \le1/4$). To this aim, we combine tools from rough paths theory and stochastic analysis.

We consider a random walk on a supercritical Galton–Watson tree with leaves, where the transition probabilities of the walk are determined by biases that are randomly assigned to the edges of the tree. The biases are chosen independently on distinct edges, each one according to a given law that satisfies a logarithmic nonlattice condition. We determine the condition under which the walk is sub-ballistic, and, in the sub-ballistic regime, we find a formula for the exponent $\gamma \in(0,1)$ such that the distance $\vert X(n)\vert$ moved by the walk in time $n$ is of the order of $n^{\gamma }$. We prove a stable limiting law for walker distance at late time, proving that the rescaled walk $n^{-\gamma }\vert X(n)\vert$ converges in distribution to an explicitly identified function of the stable law of index $\gamma $.

This paper is a counterpart to Ben Arous et al. [Ann. Probab. 40 (2012) 280–338], in which it is proved that, in the model where the biases on edges are taken to be a given constant, there is a logarithmic periodicity effect that prevents the existence of a stable limit law for scaled walker displacement. It is randomization of edge-biases that is responsible for the emergence of the stable limit in the present article, while also introducing further correlations into the model in comparison with the constant bias case. The derivation requires the development of a detailed understanding of trap geometry and the interplay between traps and backbone. The paper may be considered as a sequel to Ben Arous and Hammond [Comm. Pure Appl. Math. 65 (2012) 1481–1527], since it makes use of a result on the regular tail of the total conductance of a randomly biased subcritical Galton–Watson tree.