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Erwin Bolthausen

Publications and source records attributed to Erwin Bolthausen.

At least 19 recordsLinked to original sources
arXiv2025

Self-repellent Brownian Bridges in an Interacting Bose Gas

We consider a model of $d$-dimensional interacting quantum Bose gas, expressed in terms of an ensemble of interacting Brownian bridges in a large box and undergoing the influence of all the interactions between the legs of each of the Brownian bridges. We study the thermodynamic limit of the system and give an explicit formula for the limiting free energy and a necessary and sufficient criterion for the occurrence of a condensation phase transition. For $d\geq 5$ and sufficiently small interaction, we prove that the condensate phase is not empty. The ideas of proof rely on the similarity of the interaction to that of the self-repellent random walk, and build on a lace expansion method conducive to treating {\it paths} undergoing mutual repellence within each bridge.

math.PR↗
arXiv2021

Gardner formula for Ising perceptron models at small densities

We consider the Ising perceptron model with N spins and M = N*alpha patterns, with a general activation function U that is bounded above. For U bounded away from zero, or U a one-sided threshold function, it was shown by Talagrand (2000, 2011) that for small densities alpha, the free energy of the model converges in the large-N limit to the replica symmetric formula conjectured in the physics literature (Krauth--Mezard 1989, see also Gardner--Derrida 1988). We give a new proof of this result, which covers the more general class of all functions U that are bounded above and satisfy a certain variance bound. The proof uses the (first and second) moment method conditional on the approximate message passing iterates of the model. In order to deduce our main theorem, we also prove a new concentration result for the perceptron model in the case where U is not bounded away from zero.

math.PR↗
arXiv2018

A Morita type proof of the replica-symmetric formula for SK

We give a proof of the replica symmetric formula for the free energy of the Sherrington-Kirkpatrick model in high temperature which is based on the TAP formula. This is achieved by showing that the conditional annealed free energy equals the quenched one, where the conditioning is given by an appropriate σ-field with respect to which the TAP solutions are measurable.

math.PR↗
arXiv2017

Mean-Field interacton of Brownian occupation measures. II: A rigorous construction of the Pekar process

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].

math.PR↗
arXiv2017

Torsional rigidity for regions with a Brownian boundary

Let $T^m$ be the $m$-dimensional unit torus, $m \in N$. The torsional rigidity of an open set $Ω\subset T^m$ is the integral with respect to Lebesgue measure over all starting points $x \in Ω$ of the expected lifetime in $Ω$ of a Brownian motion starting at $x$. In this paper we consider $Ω= T^m \backslash β[0,t]$, the complement of the path $β[0,t]$ of an independent Brownian motion up to time $t$. We compute the leading order asymptotic behaviour of the expectation of the torsional rigidity in the limit as $t \to \infty$. For $m=2$ the main contribution comes from the components in $T^2 \backslash β[0,t]$ whose inradius is comparable to the largest inradius, while for $m=3$ most of $T^3 \backslash β[0,t]$ contributes. A similar result holds for $m \geq 4$ after the Brownian path is replaced by a shrinking Wiener sausage $W_{r(t)}[0,t]$ of radius $r(t)=o(t^{-1/(m-2)})$, provided the shrinking is slow enough to ensure that the torsional rigidity tends to zero. Asymptotic properties of the capacity of $β[0,t]$ in $R^3$ and $W_1[0,t]$ in $R^m$, $m \geq 4$, play a central role throughout the paper. Our results contribute to a better understanding of the geometry of the complement of Brownian motion on $T^m$, which has received a lot of attention in the literature in past years.

math.PR↗
arXiv2017

Exponential decay of covariances for the supercritical membrane model

We consider the membrane model, that is the centered Gaussian field on $\mathbb Z^d$ whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a $δ-$pinning condition, giving a reward of strength $\varepsilon$ for the field to be $0$ at any site of the lattice. In this paper we prove that in dimensions $d\geq 5$ covariances of the pinned field decay at least exponentially, as opposed to the field without pinning, where the decay is polynomial. The proof is based on estimates for certain discrete weighted norms, a percolation argument and on a Bernoulli domination result.

math.PR↗
arXiv2016

Lace expansion for dummies

We show Green's function asymptotic upper bound for the two-point function of weakly self-avoiding walk in dimension bigger than 4, revisiting a classic problem. Our proof relies on Banach algebras to analyse the lace-expansion fixed point equation and is simpler than previous approaches in that it avoids Fourier transforms.

math.PR↗
arXiv2016

Fast decay of covariances under $δ-$pinning in the critical and supercritical membrane model

We consider the membrane model, that is the centered Gaussian field on $\mathbb Z^d$ whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a $δ-$pinning condition, giving a reward of strength $\varepsilon$ for the field to be $0$ at any site of the lattice. In this paper we prove that in dimensions $d\geq 4$ covariances of the pinned field decay at least stretched-exponentially, as opposed to the field without pinning, where the decay is polynomial in $d\geq 5$ and logarithmic in $d=4.$ The proof is based on estimates for certain discrete Sobolev norms, and on a Bernoulli domination result.

math.PR↗
arXiv2015

Exit laws from large balls of (an)isotropic random walks in random environment

We study exit laws from large balls in $\mathbb{Z}^d$, $d\geq3$, of random walks in an i.i.d. random environment that is a small perturbation of the environment corresponding to simple random walk. Under a centering condition on the measure governing the environment, we prove that the exit laws are close to those of a symmetric random walk, which we identify as a perturbed simple random walk. We obtain bounds on total variation distances as well as local results comparing exit probabilities on boundary segments. As an application, we prove transience of the random walks in random environment. Our work includes the results on isotropic random walks in random environment of Bolthausen and Zeitouni [Probab. Theory Related Fields 138 (2007) 581-645]. Since several proofs in Bolthausen and Zeitouni (2007) were incomplete, a somewhat different approach was given in the first author's thesis [Long-time behavior of random walks in random environment (2013) Zürich Univ.]. Here, we extend this approach to certain anisotropic walks and provide a further step towards a fully perturbative theory of random walks in random environment.

math.PR↗
arXiv2014

Scaling limits for weakly pinned Gaussian random fields under the presence of two possible candidates

We study the scaling limit and prove the law of large numbers for weakly pinned Gaussian random fields under the critical situation that two possible candidates of the limits exist at the level of large deviation principle. This paper extends the results of [3], [7] for one dimensional fields to higher dimensions: d \geq 3, at least if the strength of pinning is sufficiently large.

math.PR↗
arXiv2014

A local CLT for convolution equations with an application to weakly self-avoiding random walks

We prove error bounds in a central limit theorem for solutions of certain convolution equations. The main motivation for investigating these equations stems from applications to lace expansions, in particular to weakly self-avoiding random walks in high dimensions. As an application we treat such self-avoiding walks in continuous space. The bounds obtained are sharper than the ones obtained by other methods.

math.PR↗
arXiv2010

Recursions and tightness for the maximum of the discrete, two dimensional Gaussian Free Field

We consider the maximum of the discrete two dimensional Gaussian free field in a box, and prove the existence of a (dense) deterministic subsequence along which the maximum, centered at its mean, is tight; this still leaves open the conjecture that tightness holds without the need for subsequences. The method of proof relies on an argument developed by Dekking and Host for branching random walks with bounded increments and on comparison results specific to Gaussian fields.

math.PR↗
arXiv2008

The quenched critical point of a diluted disordered polymer model

We consider a model for a polymer interacting with an attractive wall through a random sequence of charges. We focus on the so-called diluted limit, when the charges are very rare but have strong intensity. In this regime, we determine the quenched critical point of the model, showing that it is different from the annealed one. The proof is based on a rigorous renormalization procedure. Applications of our results to the problem of a copolymer near a selective interface are discussed.

math.PR↗
arXiv2008

Universal structures in some mean field spin glasses, and an application

We discuss a spin glass reminiscent of the Random Energy Model, which allows in particular to recast the Parisi minimization into a more classical Gibbs variational principle, thereby shedding some light on the physical meaning of the order parameter of the Parisi theory. As an application, we study the impact of an extensive cavity field on Derrida's REM: Despite its simplicity, this model displays some interesting features such as ultrametricity and chaos in temperature.

math.PR↗
arXiv2007

Lingering random walks in random environment on a strip

We consider a recurrent random walk (RW) in random environment (RE) on a strip. We prove that if the RE is i. i. d. and its distribution is not supported by an algebraic subsurface in the space of parameters defining the RE then the RW exhibits the "(log t)-squared" asymptotic behaviour. The exceptional algebraic subsurface is described by an explicit system of algebraic equations. One-dimensional walks with bounded jumps in a RE are treated as a particular case of the strip model. If the one dimensional RE is i. i. d., then our approach leads to a complete and constructive classification of possible types of asymptotic behaviour of recurrent random walks. Namely, the RW exhibits the $(\log t)^{2}$ asymptotic behaviour if the distribution of the RE is not supported by a hyperplane in the space of parameters which shall be explicitly described. And if the support of the RE belongs to this hyperplane then the corresponding RW is a martingale and its asymptotic behaviour is governed by the Central Limit Theorem.

math.PR↗
arXiv2006

Multiscale analysis of exit distributions for random walks in random environments

We present a multiscale analysis for the exit measures from large balls in Z^d, d\geq 3, of random walks in certain i.i.d. random environments which are small perturbations of the fixed environment corresponding to simple random walk. Our main assumption is an isotropy assumption on the law of the environment, introduced by Bricmont and Kupianien. The analysis is based on propagating estimates on the variational distance between the exit measure and that of simple random walk, in addition to estimates on the variational distance between smoothed versions of these quantities.

math.PR↗