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This Festschrift on the occasion of the 75th birthday of S.R.S. Varadhan, one of the most influential researchers in probability of the last fifty years, grew out of a workshop held at the Technical University of Berlin, 15–19 August, 2016. This volume contains ten research articles authored by several of Varadhan's former PhD students or close collaborators. The topics of the contributions are more or less closely linked with some of Varadhan's deepest interests over the decades: large deviations, Markov processes, interacting particle systems, motions in random media and homogenization, reaction-diffusion equations, and directed last-passage percolation.

The articles present original research on some of the most discussed current questions at the boundary between analysis and probability, with an impact on understanding phenomena in physics. This collection will be of great value to researchers with an interest in models of probability-based statistical mechanics.



Yang–Mills for Probabilists

The rigorous construction of quantum Yang–Mills theories, especially in dimension four, is one of the central open problems of mathematical physics. Construction of Euclidean Yang–Mills theories is the first step towards this goal. This article presents a formulation of some of the core aspects this problem as problems in probability theory. The presentation begins with an introduction to the basic setup of Euclidean Yang–Mills theories and lattice gauge theories. This is followed by a discussion of what is meant by a continuum limit of lattice gauge theories from the point of view of theoretical physicists. Some of the main issues are then posed as problems in probability. The article ends with a brief review of the mathematical literature.
Sourav Chatterjee

Multiscale Systems, Homogenization, and Rough Paths

In recent years, substantial progress was made towards understanding convergence of fast-slow deterministic systems to stochastic differential equations. In contrast to more classical approaches, the assumptions on the fast flow are very mild. We survey the origins of this theory and then revisit and improve the analysis of Kelly-Melbourne [Ann. Probab. Volume 44, Number 1 (2016), 479–520], taking into account recent progress in p-variation and càdlàg rough path theory.
Ilya Chevyrev, Peter K. Friz, Alexey Korepanov, Ian Melbourne, Huilin Zhang

The Deterministic and Stochastic Shallow Lake Problem

We study the welfare function of the deterministic and stochastic shallow lake problem. We show that the welfare function is the viscosity solution of the associated Bellman equation, we establish several properties including its asymptotic behaviour at infinity and we present a convergent monotone numerical scheme.
G. T. Kossioris, M. Loulakis, P. E. Souganidis

Independent Particles in a Dynamical Random Environment

We study the motion of independent particles in a dynamical random environment on the integer lattice. The environment has a product distribution. For the multidimensional case, we characterize the class of spatially ergodic invariant measures. These invariant distributions are mixtures of inhomogeneous Poisson product measures that depend on the past of the environment. We also investigate the correlations in this measure. For dimensions one and two, we prove convergence to equilibrium from spatially ergodic initial distributions. In the one-dimensional situation we study fluctuations of the net current seen by an observer traveling at a deterministic speed. When this current is centered by its quenched mean its limit distributions are the same as for classical independent particles.
Mathew Joseph, Firas Rassoul-Agha, Timo Seppäläinen

Stable Limit Laws for Reaction-Diffusion in Random Environment

We prove the emergence of stable fluctuations for reaction-diffusion in random environment with Weibull tails. This completes our work around the quenched to annealed transition phenomenon in this context of reaction diffusion. In Ben Arous et al (Transition asymptotics for reaction-diffusion in random media. Probability and mathematical physics, American Mathematical Society, Providence, RI, pp 1–40, 2007, [8]), we had already considered the model treated here and had studied fully the regimes where the law of large numbers is satisfied and where the fluctuations are Gaussian, but we had left open the regime of stable fluctuations. Our work is based on a spectral approach centered on the classical theory of rank-one perturbations. It illustrates the gradual emergence of the role of the higher peaks of the environments. This approach also allows us to give the delicate exact asymptotics of the normalizing constants needed in the stable limit law.
Gérard Ben Arous, Stanislav Molchanov, Alejandro F. Ramírez

Quenched Central Limit Theorem for the Stochastic Heat Equation in Weak Disorder

We continue with the study of the mollified stochastic heat equation in \(d\ge 3\) given by \(\mathrm{d}u_{\varepsilon ,t}=\frac{1}{2}\Delta u_{\varepsilon ,t}\mathrm{d}t+ \beta \varepsilon ^{(d-2)/2} \,u_{\varepsilon ,t} \,\mathrm{d}B_{\varepsilon ,t}\) with spatially smoothened cylindrical Wiener process B, whose (renormalized) Feynman-Kac solution describes the partition function of the continuous directed polymer. This partition function defines a (quenched) polymer path measure for every realization of the noise and we prove that as long as \(\beta >0\) stays small enough, the distribution of the diffusively rescaled Brownian path converges under the aforementioned polymer path measure to the standard Gaussian distribution.
Yannic Bröker, Chiranjib Mukherjee

GOE and Marginal Distribution via Symplectic Schur Functions

We derive Sasamoto’s Fredholm determinant formula for the Tracy-Widom GOE distribution, as well as the one-point marginal distribution of the \(\mathrm{Airy}_{2\rightarrow 1}\) process, originally derived by Borodin-Ferrari-Sasamoto, as scaling limits of point-to-line and point-to-half-line directed last passage percolation with exponentially distributed waiting times. The asymptotic analysis goes through new expressions for the last passage times in terms of integrals of (the continuous analog of) symplectic and classical Schur functions, obtained recently in [6].
Elia Bisi, Nikos Zygouras

A Large Deviations Principle for the Polar Empirical Measure in the Two-Dimensional Symmetric Simple Exclusion Process

We prove an energy estimate for the polar empirical measure of the two-dimensional symmetric simple exclusion process. We deduce from this estimate and from results in (Chang et al. in Ann Probab 32:661–691, (2004) [2]) large deviations principles for the polar empirical measure and for the occupation time of the origin.
Claudio Landim, Chih-Chung Chang, Tzong-Yow Lee

On the Growth of a Superlinear Preferential Attachment Scheme

We consider an evolving preferential attachment random graph model where at discrete times a new node is attached to an old node, selected with probability proportional to a superlinear function of its degree. For such schemes, it is known that the graph evolution condenses, that is a.s. in the limit graph there will be a single random node with infinite degree, while all others have finite degree. In this note, we establish a.s. law of large numbers type limits and fluctuation results, as \(n\uparrow \infty \), for the counts of the number of nodes with degree \(k\ge 1\) at time \(n\ge 1\). These limits rigorously verify and extend a physical picture of Krapivisky et al. (Phys Rev Lett 85:4629–4632, 2000 [16]) on how the condensation arises with respect to the degree distribution.
Sunder Sethuraman, Shankar C. Venkataramani

A Natural Probabilistic Model on the Integers and Its Relation to Dickman-Type Distributions and Buchstab’s Function

Let \(\{p_j\}_{j=1}^\infty \) denote the set of prime numbers in increasing order, let \(\Omega _N\subset \mathbb {N}\) denote the set of positive integers with no prime factor larger than \(p_N\) and let \(P_N\) denote the probability measure on \(\Omega _N\) which gives to each \(n\in \Omega _N\) a probability proportional to \(\frac{1}{n}\). This measure is in fact the distribution of the random integer \(I_N\in \Omega _N\) defined by \(I_N=\prod _{j=1}^Np_j^{X_{p_j}}\), where \(\{X_{p_j}\}_{j=1}^\infty \) are independent random variables and \(X_{p_j}\) is distributed as Geom\((1-\frac{1}{p_j})\). We show that \(\frac{\log n}{\log N}\) under \(P_N\) converges weakly to the Dickman distribution. As a corollary, we recover a classical result from multiplicative number theory—Mertens’ formula. Let \(D_{\text {nat}}(A)\) denote the natural density of \(A\subset \mathbb {N}\), if it exists, and let \(D_{\text {log-indep}}(A)=\lim _{N\rightarrow \infty }P_N(A\cap \Omega _N)\) denote the density of A arising from \(\{P_N\}_{N=1}^\infty \), if it exists. We show that the two densities coincide on a natural algebra of subsets of \(\mathbb {N}\). We also show that they do not agree on the sets of \(n^\frac{1}{s}\)-smooth numbers \(\{n\in \mathbb {N}: p^+(n)\le n^\frac{1}{s}\}\), \(s>1\), where \(p^+(n)\) denotes the largest prime divisor of n. This last consideration concerns distributions involving the Dickman function. We also consider the sets of \(n^\frac{1}{s}\)-rough numbers \(\{n\in \mathbb {N}:p^-(n)\ge n^{\frac{1}{s}}\}\), \(s>1\), where \(p^-(n)\) denotes the smallest prime divisor of n. We show that the probabilities of these sets, under the uniform distribution on \([N]=\{1,\ldots , N\}\) and under the \(P_N\)-distribution on \(\Omega _N\), have the same asymptotic decay profile as functions of s, although their rates are necessarily different. This profile involves the Buchstab function. We also prove a new representation for the Buchstab function.
Ross G. Pinsky
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