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We consider a one-dimensional classical many-body system with interaction potential of Lennard-Jones type in the thermodynamic limit at low temperature $1/\beta\in(0,\infty)$. The ground state is a periodic lattice. We show that when the density is strictly smaller than the density of the ground state lattice, the system with $N$ particles fills space by alternating approximately crystalline domains (clusters) with empty domains (voids) due to cracked bonds. The number of domains is of the order of $N\exp(- \beta e_\mathrm{surf}/2)$ with $e_\mathrm{surf}>0$ a surface energy. For the proof, the system is mapped to an effective model, which is a low-density lattice gas of defects. The results require conditions on the interactions between defects. We succeed in verifying these conditions for next-nearest neighbor interactions, applying recently derived uniform estimates of correlations., Comment: 31 pages

We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard-Jones type. The pressure (stress) is assumed small but positive and bounded away from zero, while the temperature $\beta^{-1}$ goes to zero. Our main results are: (1) As $\beta \to \infty$ at fixed positive pressure $p>0$, the Gibbs measures $\mu_\beta$ and $\nu_\beta$ for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals $\overline{\mathcal{E}}_{\mathrm{bulk}}$ and $\overline{\mathcal{E}}_\mathrm{surf}$. The minimizer of the surface functional corresponds to zero temperature boundary layers. (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of $\overline{\mathcal{E}}_\mathrm{surf}$. (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts. (4) Bounds on the decay of correlations are provided, some of them uniform in $\beta$.

We consider random Schr\"odinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowest-lying eigenvalues in the limit when the lattice spacing tends to zero. Under a suitable moment assumption on the random potential and regularity of the spatial dependence of its mean, we prove that the eigenvalues of the random operator converge to those of a deterministic Schr\"odinger operator. Assuming also regularity of the variance, the fluctuation of the random eigenvalues around their mean are shown to obey a multivariate central limit theorem. This extends the authors' recent work where similar conclusions have been obtained for bounded random potentials., Comment: 25 pages

We study the solutions $u=u(x,t)$ to the Cauchy problem on $\mathbb Z^d\times(0,\infty)$ for the parabolic equation $\partial_t u=\Delta u+\xi u$ with initial data $u(x,0)=1_{\{0\}}(x)$. Here $\Delta$ is the discrete Laplacian on $\mathbb Z^d$ and $\xi=(\xi(z))_{z\in\mathbb Z^d}$ is an i.i.d.\ random field with doubly-exponential upper tails. We prove that, for large $t$ and with large probability, a majority of the total mass $U(t):=\sum_x u(x,t)$ of the solution resides in a bounded neighborhood of a site $Z_t$ that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian $\Delta+\xi$ and the distance to the origin. The processes $t\mapsto Z_t$ and $t \mapsto \tfrac1t \log U(t)$ are shown to converge in distribution under suitable scaling of space and time. Aging results for $Z_t$, as well as for the solution to the parabolic problem, are also established. The proof uses the characterization of eigenvalue order statistics for $\Delta+\xi$ in large sets recently proved by the first two authors., Comment: 69 pages, version to appear in Prob. Theory Rel. Fields

We consider mean-field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self-attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan [6] in terms of the {\it{Pekar variational formula}}, which coincides with the behavior of the partition function of the {\it{polaron problem}} under strong coupling. Based on this, in 1986 Spohn [14] made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the {\it{Pekar process}}, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean-field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean-field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence, is a contribution to the understanding of the "mean-field approximation" of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed in [11], its extension to the uniform strong metric for the singular Coulomb interaction carried out in [8], as well as an idea inspired by a {\it{partial path exchange}} argument appearing in [1]., Comment: Following referee's suggestion, Section 4 is expanded and Proof of Theorem 2.1 is corrected therein. To appear in "Comm. Pure. Appl. Math."

We study the statistics of Dirichlet eigenvalues of the random Schr\"odinger operator $-\epsilon^{-2}\Delta^{(\text{d})}+\xi^{(\epsilon)}(x)$, with $\Delta^{(\text{d})}$ the discrete Laplacian on $\mathbb Z^d$ and $\xi^{(\epsilon)}(x)$ uniformly bounded independent random variables, on sets of the form $D_\epsilon:=\{x\in \mathbb Z^d\colon x\epsilon\in D\}$ for $D\subset \mathbb R^d$ bounded, open and with a smooth boundary. If $\mathbb E\xi^{(\epsilon)}(x)=U(x\epsilon)$ holds for some bounded and continuous $U\colon D\to \mathbb R$, we show that, as $\epsilon\downarrow0$, the $k$-th eigenvalue converges to the $k$-th Dirichlet eigenvalue of the homogenized operator $-\Delta+U(x)$, where $\Delta$ is the continuum Dirichlet Laplacian on $D$. Assuming further that $\text{Var}(\xi^{(\epsilon)}(x))=V(x\epsilon)$ for some positive and continuous $V\colon D\to \mathbb R$, we establish a multivariate central limit theorem for simple eigenvalues centered by their expectation. The limiting covariance for a given pair of simple eigenvalues is expressed as an integral of $V$ against the product of squares of the corresponding eigenfunctions of $-\Delta+U(x)$., Comment: 26 pages, to appear in SIAM J. Math. Anal

We consider random Schr\"odinger operators of the form $\Delta+\xi$, where $\Delta$ is the lattice Laplacian on $\mathbb Z^d$ and $\xi$ is an i.i.d. random field, and study the extreme order statistics of the eigenvalues for this operator restricted to large but finite subsets of $\mathbb Z^d$. We show that for $\xi$ with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class. The corresponding eigenfunctions are exponentially localized in regions where $\xi$ takes large, and properly arranged, values. A new and self-contained argument is thus provided for Anderson localization at the spectral edge which permits a rather explicit description of the shape of the potential and the eigenfunctions. Our study serves as an input into the analysis of an associated parabolic Anderson problem., Comment: 36 pages

An interesting problem in statistical physics is the condensation of classical particles in droplets or clusters when the pair-interaction is given by a stable Lennard-Jones-type potential. We study two aspects of this problem. We start by deriving a large deviations principle for the cluster size distribution for any inverse temperature $\beta\in (0,\infty)$ and particle density $\rho\in(0,\rho_{\mathrm{cp}})$ in the thermodynamic limit. Here $\rho_{\mathrm{cp}}>0$ is the close packing density. While in general the rate function is an abstract object, our second main result is the $\Gamma$-convergence of the rate function toward an explicit limiting rate function in the low-temperature dilute limit $\beta\to \infty$, $\rho\downarrow0$ such that $-\beta^{-1}\log\rho\to\nu$ for some $\nu\in(0,\infty)$. The limiting rate function and its minimisers appeared in recent work, where the temperature and the particle density were coupled with the particle number. In the decoupled limit considered here, we prove that just one cluster size is dominant, depending on the parameter $\nu$. Under additional assumptions on the potential, the $\Gamma$-convergence along curves can be strengthened to uniform bounds, valid in a low-temperature, low-density rectangle., Comment: Published in at http://dx.doi.org/10.1214/14-AAP1014 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider $N$ bosons in a box in $\mathbb {R}^d$ with volume $N/\rho$ under the influence of a mutually repellent pair potential. The particle density $\rho\in (0,\infty)$ is kept fixed. Our main result is the identification of the limiting free energy, $f(\beta,\rho)$, at positive temperature $1/\beta$, in terms of an explicit variational formula, for any fixed $\rho$ if $\beta$ is sufficiently small, and for any fixed $\beta$ if $\rho$ is sufficiently small. The thermodynamic equilibrium is described by the symmetrized trace of $e^{-\beta {\mathcal{H}}_N}$, where ${\mathcal{H}}_N$ denotes the corresponding Hamilton operator. The well-known Feynman--Kac formula reformulates this trace in terms of $N$ interacting Brownian bridges. Due to the symmetrization, the bridges are organized in an ensemble of cycles of various lengths. The novelty of our approach is a description in terms of a marked Poisson point process whose marks are the cycles. This allows for an asymptotic analysis of the system via a large-deviations analysis of the stationary empirical field. The resulting variational formula ranges over random shift-invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process. In our proof of the lower bound for the free energy, we drop all interaction involving "infinitely long" cycles, and their possible presence is signalled by a loss of mass of the "finitely long" cycles in the variational formula. In the proof of the upper bound, we only keep the mass on the "finitely long" cycles. We expect that the precise relationship between these two bounds lies at the heart of Bose--Einstein condensation and intend to analyze it further in future., Comment: Published in at http://dx.doi.org/10.1214/10-AOP565 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We review probabilistic approaches to the Gross-Pitaevskii theory describing interacting dilute systems of particles. The main achievement are large deviations principles for the mean occupation measure of a large system of interacting Brownian motions in a trapping potential. The corresponding rate functions are given as variational problems whose solution provide effective descriptions of the infinite system.

We investigate the local times of a continuous-time Markov chain on an arbitrary discrete state space. For fixed finite range of the Markov chain, we derive an explicit formula for the joint density of all local times on the range, at any fixed time. We use standard tools from the theory of stochastic processes and finite-dimensional complex calculus. We apply this formula in the following directions: (1) we derive large deviation upper estimates for the normalized local times beyond the exponential scale, (2) we derive the upper bound in Varadhan's lemma for any measurable functional of the local times, and (3) we derive large deviation upper bounds for continuous-time simple random walk on large subboxes of $\mathbb{Z}^d$ tending to $\mathbb{Z}^d$ as time diverges. We finally discuss the relation of our density formula to the Ray--Knight theorem for continuous-time simple random walk on $\mathbb{Z}$, which is analogous to the well-known Ray--Knight description of Brownian local times., Comment: Published at http://dx.doi.org/10.1214/009171906000001024 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ with some non-degenerate initial measure on some fixed time interval $[0,\beta]$ with symmetrised initial-terminal condition. That is, for any $i$, the terminal location of the $i$-th motion is affixed to the initial point of the $\sigma(i)$-th motion, where $\sigma$ is a uniformly distributed random permutation of $1,...,N$. Such systems play an important role in quantum physics in the description of Boson systems at positive temperature $1/\beta$. In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Dirac measures in the $N$ paths) and of the mean of the normalised occupation measures of the $N$ motions in terms of large deviations principles. The rate functions are given as variational formulas involving certain entropies and Fenchel-Legendre transforms. Consequences are drawn for asymptotic independence statements and laws of large numbers. In the special case related to quantum physics, our rate function for the occupation measures turns out to be equal to the well-known Donsker-Varadhan rate function for the occupation measures of one motion in the limit of diverging time. This enables us to prove a simple formula for the large-N asymptotic of the symmetrised trace of ${\rm e}^{-\beta \mathcal{H}_N}$, where $\mathcal{H}_N$ is an $N$-particle Hamilton operator in a trap.

We study a model of $ N $ mutually repellent Brownian motions under confinement to stay in some bounded region of space. Our model is defined in terms of a transformed path measure under a trap Hamiltonian, which prevents the motions from escaping to infinity, and a pair-interaction Hamiltonian, which imposes a repellency of the $N$ paths. In fact, this interaction is an $N$-dependent regularisation of the Brownian intersection local times, an object which is of independent interest in the theory of stochastic processes. The time horizon (interpreted as the inverse temperature) is kept fixed. We analyse the model for diverging number of Brownian motions in terms of a large deviation principle. The resulting variational formula is the positive-temperature analogue of the well-known Gross-Pitaevskii formula, which approximates the ground state of a certain dilute large quantum system; the kinetic energy term of that formula is replaced by a probabilistic energy functional. This study is a continuation of the analysis in \cite{ABK04} where we considered the limit of diverging time (i.e., the zero-temperature limit) with fixed number of Brownian motions, followed by the limit for diverging number of motions. \bibitem[ABK04]{ABK04} {\sc S.~Adams, J.-B.~Bru} and {\sc W.~K\"onig}, \newblock Large deviations for trapped interacting Brownian particles and paths, \newblock {\it Ann. Probab.}, to appear (2004).

We discuss the long time behaviour of the parabolic Anderson model, the Cauchy problem for the heat equation with random potential on $\Z^d$. We consider general i.i.d. potentials and show that exactly \emph{four} qualitatively different types of intermittent behaviour can occur. These four universality classes depend on the upper tail of the potential distribution: (1) tails at $\infty$ that are thicker than the double-exponential tails, (2) double-exponential tails at $\infty$ studied by G\"artner and Molchanov, (3) a new class called \emph{almost bounded potentials}, and (4) potentials bounded from above studied by Biskup and K\"onig. The new class (3), which contains both unbounded and bounded potentials, is studied in both the annealed and the quenched setting. We show that intermittency occurs on unboundedly increasing islands whose diameter is slowly varying in time. The characteristic variational formulas describing the optimal profiles of the potential and of the solution are solved explicitly by parabolas, respectively, Gaussian densities., Comment: 39 pages

Let (Z_n)_{n\in\N_0} be a d-dimensional random walk in random scenery, i.e., Z_n=\sum_{k=0}^{n-1}Y_{S_k} with (S_k)_{k\in\N_0} a random walk in Z^d and (Y_z)_{z\in Z^d} an i.i.d. scenery, independent of the walk. We assume that the random variables Y_z have a stretched exponential tail. In particular, they do not possess exponential moments. We identify the speed and the rate of the logarithmic decay of Pr(Z_n>t_n n) for all sequences (t_n)_{n\in\N} satisfying a certain lower bound. This complements previous results, where it was assumed that Y_z has exponential moments of all orders. In contrast to the previous situation,the event \{Z_n>t_nn\} is not realized by a homogeneous behavior of the walk's local times and the scenery, but by many visits of the walker to a particular site and a large value of the scenery at that site. This reflects a well-known extreme behavior typical for random variables having no exponential moments., Comment: 12 pages

We consider the large-time behavior of the solution $u\colon [0,\infty)\times\Z\to[0,\infty)$ to the parabolic Anderson problem $\partial_t u=\kappa\Delta u+\xi u$ with initial data $u(0,\cdot)=1$ and non-positive finite i.i.d. potentials $(\xi(z))_{z\in\Z}$. Unlike in dimensions $d\ge2$, the almost-sure decay rate of $u(t,0)$ as $t\to\infty$ is not determined solely by the upper tails of $\xi(0)$; too heavy lower tails of $\xi(0)$ accelerate the decay. The interpretation is that sites $x$ with large negative $\xi(x)$ hamper the mass flow and hence screen off the influence of more favorable regions of the potential. The phenomenon is unique to $d=1$. The result answers an open question from our previous study \cite{BK00} of this model in general dimension., Comment: LaTeX+times package, 14 pages

We consider the parabolic Anderson problem $\partial_t u=\kappa\Delta u+\xi u$ on $(0,\infty)\times \Z^d$ with random i.i.d. potential $\xi=(\xi(z))_{z\in\Z^d}$ and the initial condition $u(0,\cdot)\equiv1$. Our main assumption is that $\esssup\xi(0)=0$. Depending on the thickness of the distribution $\prob(\xi(0)\in\cdot)$ close to its essential supremum, we identify both the asymptotics of the moments of $u(t,0)$ and the almost-sure asymptotics of $u(t,0)$ as $t\to\infty$ in terms of variational problems. As a by-product, we establish Lifshitz tails for the random Schr\"odinger operator $-\kappa\Delta-\xi$ at the bottom of its spectrum. In our class of $\xi$ distributions, the Lifshitz exponent ranges from $d/2$ to $\infty$; the power law is typically accompanied by lower-order corrections., Comment: 40 pages, LaTeX 2e+times, version published in Ann. Probab