Large deviations for near-extreme eigenvalues in the beta-ensembles

For beta ensembles with convex poynomial potentials, we prove a large deviation principle for the empirical spectral distribution seen from the rightmost particle. This modified spectral distribution was introduced by Perret and Schehr (J. Stat. Phys. 2014) to study the crowding near the maximal eigenvalue, in the case of the GUE. We prove also convergence of fluctuations.Comment: We fixed typos and changed Remarks 2.13 and 2.1

A large deviation principle for empirical measures on Polish spaces: Application to singular Gibbs measures on manifolds

We prove a large deviation principle for a sequence of point processes defined by Gibbs probability measures on a Polish space. This is obtained as a consequence of a more general Laplace principle for the non-normalized Gibbs measures. We consider three main applications: Conditional Gibbs measures on compact spaces, Coulomb gases on compact Riemannian manifolds and the usual Gibbs measures in the Euclidean space. Finally, we study the generalization of Fekete points and prove a deterministic version of the Laplace principle known as $\Gamma$-convergence. The approach is partly inspired by the works of Dupuis and co-authors. It is remarkably natural and general compared to the usual strategies for singular Gibbs measures.Comment: 23 pages, abstract also in french, for more details in the proofs see version 1, application to Gaussian polynomials adde

Large deviations for a random speed particle

We investigate large deviations for the empirical measure of the position and momentum of a particle traveling in a box with hot walls. The particle travels with uniform speed from left to right, until it hits the right boundary. Then it is absorbed and re-emitted from the left boundary with a new random speed, taken from an i.i.d. sequence. It turns out that this simple model, often used to simulate a heat bath, displays unusually complex large deviations features, that we explain in detail. In particular, if the tail of the update distribution of the speed is sufficiently oscillating, then the empirical measure does not satisfy a large deviations principle, and we exhibit optimal lower and upper large deviations functionals

The Large Deviation Principle for Coarse-Grained Processes

The large deviation principle is proved for a class of $L^2$-valued processes that arise from the coarse-graining of a random field. Coarse-grained processes of this kind form the basis of the analysis of local mean-field models in statistical mechanics by exploiting the long-range nature of the interaction function defining such models. In particular, the large deviation principle is used in a companion paper to derive the variational principles that characterize equilibrium macrostates in statistical models of two-dimensional and quasi-geostrophic turbulence. Such macrostates correspond to large-scale, long-lived flow structures, the description of which is the goal of the statistical equilibrium theory of turbulence. The large deviation bounds for the coarse-grained process under consideration are shown to hold with respect to the strong $L^2$ topology, while the associated rate function is proved to have compact level sets with respect to the weak topology. This compactness property is nevertheless sufficient to establish the existence of equilibrium macrostates for both the microcanonical and canonical ensembles.Comment: 19 page

A Renewal version of the Sanov theorem

Large deviations for the local time of a process $X_t$ are investigated, where $X_t=x_i$ for $t \in [S_{i-1},S_i[$ and $(x_j)$ are i.i.d.\ random variables on a Polish space, $S_j$ is the $j$-th arrival time of a renewal process depending on $(x_j)$. No moment conditions are assumed on the arrival times of the renewal process.Comment: 13 page

Process-level quenched large deviations for random walk in random environment

We consider a bounded step size random walk in an ergodic random environment with some ellipticity, on an integer lattice of arbitrary dimension. We prove a level 3 large deviation principle, under almost every environment, with rate function related to a relative entropy.Comment: Proof of (6.2) corrected. Lemma A.2 replace

A large deviation approach to optimal transport

A probabilistic method for solving the Monge-Kantorovich mass transport problem on $R^d$ is introduced. A system of empirical measures of independent particles is built in such a way that it obeys a doubly indexed large deviation principle with an optimal transport cost as its rate function. As a consequence, new approximation results for the optimal cost function and the optimal transport plans are derived. They follow from the Gamma-convergence of a sequence of normalized relative entropies toward the optimal transport cost. A wide class of cost functions including the standard power cost functions $|x-y|^p$ enter this framework

Large deviations for random walks in a random environment on a strip

We consider a random walk in a random environment (RWRE) on the strip of finite width $\mathbb{Z} \times \{1,2,\ldots,d\}$. We prove both quenched and averaged large deviation principles for the position and the hitting times of the RWRE. Moreover, we prove a variational formula that relates the quenched and averaged rate functions, thus extending a result of Comets, Gantert, and Zeitouni for nearest-neighbor RWRE on $\mathbb{Z}