Sub-Gaussian short time asymptotics for measure metric Dirichlet spaces

This paper presents estimates for the distribution of the exit time from balls and short time asymptotics for measure metric Dirichlet spaces. The estimates cover the classical Gaussian case, the sub-diffusive case which can be observed on particular fractals and further less regular cases as well. The proof is based on a new chaining argument and it is free of volume growth assumptions

Estimating the spectrum of a density operator

Given N quantum systems prepared according to the same density operator \rho, we propose a measurement on the N-fold system which approximately yields the spectrum of \rho. The projections of the proposed observable decompose the Hilbert space according to the irreducible representations of the permutations on N points, and are labeled by Young frames, whose relative row lengths estimate the eigenvalues of \rho in decreasing order. We show convergence of these estimates in the limit N\to\infty, and that the probability for errors decreases exponentially with a rate we compute explicitly.Comment: 4 Pages, RevTeX, one figur

Long Cycles in a Perturbed Mean Field Model of a Boson Gas

In this paper we give a precise mathematical formulation of the relation between Bose condensation and long cycles and prove its validity for the perturbed mean field model of a Bose gas. We decompose the total density $\rho=\rho_{{\rm short}}+\rho_{{\rm long}}$ into the number density of particles belonging to cycles of finite length ($\rho_{{\rm short}}$) and to infinitely long cycles ($\rho_{{\rm long}}$) in the thermodynamic limit. For this model we prove that when there is Bose condensation, $\rho_{{\rm long}}$ is different from zero and identical to the condensate density. This is achieved through an application of the theory of large deviations. We discuss the possible equivalence of $\rho_{{\rm long}}\neq 0$ with off-diagonal long range order and winding paths that occur in the path integral representation of the Bose gas.Comment: 10 page

Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise

The Freidlin-Wentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. As examples, the main results are applied to derive the large deviation principle for different types of SPDE such as stochastic reaction-diffusion equations, stochastic porous media equations and fast diffusion equations, and the stochastic p-Laplace equation in Hilbert space. The weak convergence approach is employed in the proof to establish the Laplace principle, which is equivalent to the large deviation principle in our framework.Comment: 31 pages, published in Appl. Math. Opti

Equality of averaged and quenched large deviations for random walks in random environments in dimensions four and higher

We consider large deviations for nearest-neighbor random walk in a uniformly elliptic i.i.d. environment. It is easy to see that the quenched and the averaged rate functions are not identically equal. When the dimension is at least four and Sznitman's transience condition (T) is satisfied, we prove that these rate functions are finite and equal on a closed set whose interior contains every nonzero velocity at which the rate functions vanish.Comment: 17 pages. Minor revision. In particular, note the change in the title of the paper. To appear in Probability Theory and Related Fields

Equilibrium fluctuations of additive functionals of zero-range models

For mean-zero and asymmetric zero-range processes on $\Z^d$, the fluctuations of additive functionals starting from an invariant measure are considered. Under certain assumptions, we establish when the fluctuations are diffusive and satisfy functional central limit theorems. These results complement those for symmetric zero-range systems and also those for simple exclusion models already in the literature.FC

The Continuum Directed Random Polymer

Motivated by discrete directed polymers in one space and one time dimension, we construct a continuum directed random polymer that is modeled by a continuous path interacting with a space-time white noise. The strength of the interaction is determined by an inverse temperature parameter beta, and for a given beta and realization of the noise the path evolves in a Markovian way. The transition probabilities are determined by solutions to the one-dimensional stochastic heat equation. We show that for all beta > 0 and for almost all realizations of the white noise the path measure has the same Holder continuity and quadratic variation properties as Brownian motion, but that it is actually singular with respect to the standard Wiener measure on C([0,1]).Comment: 21 page

Lattice gas model in random medium and open boundaries: hydrodynamic and relaxation to the steady state

We consider a lattice gas interacting by the exclusion rule in the presence of a random field given by i.i.d. bounded random variables in a bounded domain in contact with particles reservoir at different densities. We show, in dimensions $d \ge 3$, that the rescaled empirical density field almost surely, with respect to the random field, converges to the unique weak solution of a non linear parabolic equation having the diffusion matrix determined by the statistical properties of the external random field and boundary conditions determined by the density of the reservoir. Further we show that the rescaled empirical density field, in the stationary regime, almost surely with respect to the random field, converges to the solution of the associated stationary transport equation

Self-avoiding fractional Brownian motion - The Edwards model

In this work we extend Varadhan's construction of the Edwards polymer model to the case of fractional Brownian motions in $\R^d$, for any dimension $d\geq 2$, with arbitrary Hurst parameters $H\leq 1/d$.Comment: 14 page

Linear Statistics of Point Processes via Orthogonal Polynomials

For arbitrary $\beta > 0$, we use the orthogonal polynomials techniques developed by R. Killip and I. Nenciu to study certain linear statistics associated with the circular and Jacobi $\beta$ ensembles. We identify the distribution of these statistics then prove a joint central limit theorem. In the circular case, similar statements have been proved using different methods by a number of authors. In the Jacobi case these results are new.Comment: Added references, corrected typos. To appear, J. Stat. Phy