We study the exit path from a general domain after the last visit to a set of a Markov chain with rare transitions. We prove several large deviation principles for the law of the succession of the cycles visited by the process (the cycle path), the succession of the saddle points gone through to jump from cycle to cycle on the cycle path (the saddle path) and the succession of all the points gone through (the exit path). We estimate the time the process spends in each cycle of the cycle path...

The goal of this paper is to analyse the asymptotic behaviour of the cycle process and the total number of cycles of weighted and generalized weighted random permutations which are relevant models in physics and which extend the Ewens measure. We combine tools from combinatorics and complex analysis (e.g. singularity analysis of generating functions) to prove that under some analytic conditions (on relevant generating functions) the cycle process converges to a vector of independent Poisson variables...

We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the – suitably centered – empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov, see Comm. Math. Phys.92 (1984) 491–506. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the...

We study the large deviation principle for stochastic processes of the form $\{{\sum }_{k=1}^{\infty }{x}_k\left(t\right){\xi }_k:t\in T\}$, where ${\left\{{\xi }_k\right\}}_{k=1}^{\infty }$ is a sequence of i.i.d.r.v.'s with mean zero and $x_k\left(t\right)\in ℝ$. We present necessary and sufficient conditions for the large deviation principle for these stochastic processes in several situations. Our approach is based in showing the large deviation principle of the finite dimensional distributions and an exponential asymptotic equicontinuity condition. In order to get the exponential asymptotic equicontinuity condition,...

We study the large deviation principle for stochastic processes of the form $\{{\sum }_{k=1}^{\infty }{x}_k\left(t\right){\xi }_k:t\in T\}$, where ${\left\{{\xi }_k\right\}}_{k=1}^{\infty }$ is a sequence of i.i.d.r.v.’s with mean zero and $x_k\left(t\right)\in ℝ$. We present necessary and sufficient conditions for the large deviation principle for these stochastic processes in several situations. Our approach is based in showing the large deviation principle of the finite dimensional distributions and an exponential asymptotic equicontinuity condition. In order to get the exponential asymptotic equicontinuity condition,...

In this paper we study the parabolic Anderson equation $\partial u(x,t)/\partial t=\kappa 𝛥u(x,t)+\xi (x,t)u(x,t)$, $x\in {\mathbb{Z}}^d$, $t\ge 0$, where the $u$-field and the $\xi$-field are $ℝ$-valued, $\kappa \in [0,\infty )$ is the diffusion constant, and $𝛥$ is the discrete Laplacian. The $\xi$-field plays the role of adynamic random environmentthat drives the equation. The initial condition $u(x,0)=u_0\left(x\right)$, $x\in {\mathbb{Z}}^d$, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump...

The Tracy–Widom $\beta$ distribution is the large dimensional limit of the top eigenvalue of $\beta$ random matrix ensembles. We use the stochastic Airy operator representation to show that as $a\to \infty$ the tail of the Tracy–Widom distribution satisfies $$P({\mathrm{𝑇𝑊}}_{\beta }gt;a)=a^{-(3/4)\beta +\mathrm{o}\left(1\right)}exp\left(-\frac{2}{3}\beta a^{3/2}\right).$$

For a set A ⊂ C[0, ∞), we give new results on the growth of the number of particles in a branching Brownian motion whose paths fall within A. We show that it is possible to work without rescaling the paths. We give large deviations probabilities as well as a more sophisticated proof of a result on growth in the number of particles along certain sets of paths. Our results reveal that the number of particles can oscillate dramatically. We also obtain new results on the number of particles near the...

We prove that for s < 0, s-concave measures on ℝⁿ exhibit thin-shell concentration similar to the log-concave case. This leads to a Berry-Esseen type estimate for most of their one-dimensional marginal distributions. We also establish sharp reverse Hölder inequalities for s-concave measures.