A Local limit theorem for directed polymers in random media: the continuous and the discrete case

In this article, we consider two models of directed polymers in random environment: a discrete model and a continuous model. We consider these models in dimension greater or equal to 3 and we suppose that the normalized partition function is bounded in L^2. Under these assumptions, Sinai proved a local limit theorem for the discrete model, using a perturbation expansion. In this article, we give a new method for proving Sinai's local limit theorem. This new method can be transposed to the continuous setting in which we prove a similar local limit theorem

Scaling limits for symmetric Ito-Levy processes in random medium

We are concerned with scaling limits of the solutions to stochastic differential equations with stationary coefficients driven by Poisson random measures and Brownian motions. We state an annealed convergence theorem, in which the limit exhibits a diffusive or superdiffusive behavior, depending on the integrability properties of the Poisson random measureComment: 33 page

The Tail expansion of Gaussian multiplicative chaos and the Liouville reflection coefficient

In this short note, we derive a precise tail expansion for Gaussian multiplicative chaos (GMC) associated to the 2d GFF on the unit disk with zero average on the unit circle (and variants). More specifically, we show that to first order the tail is a constant times an inverse power with an explicit value for the tail exponent as well as an explicit value for the constant in front of the inverse power; we also provide a second order bound for the tail expansion. The main interest of our work consists of two points. First, our derivation is based on a simple method which we believe is universal in the sense that it can be generalized to all dimensions and to all log-correlated fields. Second, in the 2d case we consider, the value of the constant in front of the inverse power is (up to explicit terms) nothing but the Liouville reflection coefficient taken at a special value. The explicit computation of the constant was performed in the recent rigorous derivation with A. Kupiainen of the DOZZ formula \cite{KRV1,KRV}; to our knowledge, it is the first time one derives rigorously an explicit value for such a constant in the tail expansion of a GMC measure. We have deliberately kept this paper short to emphasize the method so that it becomes an easily accessible toolbox for computing tails in GMC theory.Comment: The new version contains a more general statement. We also detail the relation between the Liouville reflection coefficient and the quantum spheres introduced by Duplantier-Miller-Sheffiel

Liouville Brownian motion at criticality

In this paper, we construct the Brownian motion of Liouville Quantum Gravity with central charge $c=1$ (more precisely we restrict to the corresponding free field theory). Liouville quantum gravity with $c=1$ corresponds to two-dimensional string theory and is the conjectural scaling limit of large planar maps weighted with a $O(n=2)$ loop model or a $Q=4$-state Potts model embedded in a two dimensional surface in a conformal manner. Following \cite{GRV1}, we start by constructing the critical LBM from one fixed point $x\in\mathbb{R}^2$ (or $x\in\S^2$), which amounts to changing the speed of a standard planar Brownian motion depending on the local behaviour of the critical Liouville measure $M'(dx)=-X(x)e^{2X(x)}\,dx$ (where $X$ is a Gaussian Free Field, say on $\mathbb{S}^2$). Extending this construction simultaneously to all points in $\mathbb{R}^2$ requires a fine analysis of the potential properties of the measure $M'$. This allows us to construct a strong Markov process with continuous sample paths living on the support of $M'$, namely a dense set of Hausdorff dimension $0$. We finally construct the associated Liouville semigroup, resolvent, Green function, heat kernel and Dirichlet form. In passing, we extend to quite a general setting the construction of the critical Gaussian multiplicative chaos that was initiated in \cite{Rnew7,Rnew12} and also establish new capacity estimates for the critical Gaussian multiplicative chaos.Comment: 52 page