Intermittency on catalysts

The present paper provides an overview of results obtained in four recent papers by the authors. These papers address the problem of intermittency for the Parabolic Anderson Model in a \emph{time-dependent random medium}, describing the evolution of a ``reactant'' in the presence of a ``catalyst''. Three examples of catalysts are considered: (1) independent simple random walks; (2) symmetric exclusion process; (3) symmetric voter model. The focus is on the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of the reactant. It turns out that these exponents exhibit an interesting dependence on the dimension and on the diffusion constant.Comment: 11 pages, invited paper to appear in a Festschrift in honour of Heinrich von Weizs\"acker, on the occasion of his 60th birthday, to be published by Cambridge University Pres

Intermittency on catalysts: three-dimensional simple symmetric exclusion

We continue our study of intermittency for the parabolic Anderson model $\partial u/\partial t = \kappa\Delta u + \xi u$ in a space-time random medium $\xi$, where $\kappa$ is a positive diffusion constant, $\Delta$ is the lattice Laplacian on $\Z^d$, $d \geq 1$, and $\xi$ is a simple symmetric exclusion process on $\Z^d$ in Bernoulli equilibrium. This model describes the evolution of a \emph{reactant} $u$ under the influence of a \emph{catalyst} $\xi$. In G\"artner, den Hollander and Maillard (2007) we investigated the behavior of the annealed Lyapunov exponents, i.e., the exponential growth rates as $t\to\infty$ of the successive moments of the solution $u$. This led to an almost complete picture of intermittency as a function of $d$ and $\kappa$. In the present paper we finish our study by focussing on the asymptotics of the Lyaponov exponents as $\kappa\to\infty$ in the \emph{critical} dimension $d=3$, which was left open in G\"artner, den Hollander and Maillard (2007) and which is the most challenging. We show that, interestingly, this asymptotics is characterized not only by a \emph{Green} term, as in $d\geq 4$, but also by a \emph{polaron} term. The presence of the latter implies intermittency of \emph{all} orders above a finite threshold for $\kappa$.Comment: 38 page

Intermittency on catalysts: Voter model

In this paper we study intermittency for the parabolic Anderson equation $\partial u/\partial t=\kappa\Delta u+\gamma\xi u$ with $u:\mathbb{Z}^d\times[0,\infty)\to\mathbb{R}$, where $\kappa\in[0,\infty)$ is the diffusion constant, $\Delta$ is the discrete Laplacian, $\gamma\in(0,\infty)$ is the coupling constant, and $\xi:\mathbb{Z}^d\times[0,\infty)\to\mathbb{R}$ is a space--time random medium. The solution of this equation describes the evolution of a ``reactant'' $u$ under the influence of a ``catalyst'' $\xi$. We focus on the case where $\xi$ is the voter model with opinions 0 and 1 that are updated according to a random walk transition kernel, starting from either the Bernoulli measure $\nu_{\rho}$ or the equilibrium measure $\mu_{\rho}$, where $\rho\in(0,1)$ is the density of 1's. We consider the annealed Lyapunov exponents, that is, the exponential growth rates of the successive moments of $u$. We show that if the random walk transition kernel has zero mean and finite variance, then these exponents are trivial for $1\leq d\leq4$, but display an interesting dependence on the diffusion constant $\kappa$ for $d\geq 5$, with qualitatively different behavior in different dimensions. In earlier work we considered the case where $\xi$ is a field of independent simple random walks in a Poisson equilibrium, respectively, a symmetric exclusion process in a Bernoulli equilibrium, which are both reversible dynamics. In the present work a main obstacle is the nonreversibility of the voter model dynamics, since this precludes the application of spectral techniques. The duality with coalescing random walks is key to our analysis, and leads to a representation formula for the Lyapunov exponents that allows for the application of large deviation estimates.Comment: Published in at http://dx.doi.org/10.1214/10-AOP535 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity

We show that the n-fold integrals $\chi^{(n)}$ of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the "Ising class", or n-fold integrals from enumerative combinatorics, like lattice Green functions, are actually diagonals of rational functions. As a consequence, the power series expansions of these solutions of linear differential equations "Derived From Geometry" are globally bounded, which means that, after just one rescaling of the expansion variable, they can be cast into series expansions with integer coefficients. Besides, in a more enumerative combinatorics context, we show that generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions. We give a large set of results illustrating the fact that the unique analytical solution of Calabi-Yau ODEs, and more generally of MUM ODEs, is, almost always, diagonal of rational functions. We revisit Christol's conjecture that globally bounded series of G-operators are necessarily diagonals of rational functions. We provide a large set of examples of globally bounded series, or series with integer coefficients, associated with modular forms, or Hadamard product of modular forms, or associated with Calabi-Yau ODEs, underlying the concept of modularity. We finally address the question of the relations between the notion of integrality (series with integer coefficients, or, more generally, globally bounded series) and the modularity (in particular integrality of the Taylor coefficients of mirror map), introducing new representations of Yukawa couplings.Comment: 100 page

Integrable mappings and polynomial growth

We describe birational representations of discrete groups generated by involutions, having their origin in the theory of exactly solvable vertex-models in lattice statistical mechanics. These involutions correspond respectively to two kinds of transformations on $q \times q$ matrices: the inversion of the $q \times q$ matrix and an (involutive) permutation of the entries of the matrix. We concentrate on the case where these permutations are elementary transpositions of two entries. In this case the birational transformations fall into six different classes. For each class we analyze the factorization properties of the iteration of these transformations. These factorization properties enable to define some canonical homogeneous polynomials associated with these factorization properties. Some mappings yield a polynomial growth of the complexity of the iterations. For three classes the successive iterates, for $q=4$, actually lie on elliptic curves. This analysis also provides examples of integrable mappings in arbitrary dimension, even infinite. Moreover, for two classes, the homogeneous polynomials are shown to satisfy non trivial non-linear recurrences. The relations between factorizations of the iterations, the existence of recurrences on one or several variables, as well as the integrability of the mappings are analyzed.Comment: 45 page

A functional central limit theorem for regenerative chains

Using the regenerative scheme of Comets, Fern\'andez and Ferrari (2002), we establish a functional central limit theorem (FCLT) for discrete time stochastic processes (chains) with summable memory decay. Furthermore, under stronger assumptions on the memory decay, we identify the limiting variance in terms of the process only. As applications, we define classes of binary autoregressive processes and power-law Ising chains for which the FCLT is fulfilled.Comment: 14 page

Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations

We give the exact expressions of the partial susceptibilities $\chi^{(3)}_d$ and $\chi^{(4)}_d$ for the diagonal susceptibility of the Ising model in terms of modular forms and Calabi-Yau ODEs, and more specifically, $_3F_2([1/3,2/3,3/2],\, [1,1];\, z)$ and $_4F_3([1/2,1/2,1/2,1/2],\, [1,1,1]; \, z)$ hypergeometric functions. By solving the connection problems we analytically compute the behavior at all finite singular points for $\chi^{(3)}_d$ and $\chi^{(4)}_d$. We also give new results for $\chi^{(5)}_d$. We see in particular, the emergence of a remarkable order-six operator, which is such that its symmetric square has a rational solution. These new exact results indicate that the linear differential operators occurring in the $n$-fold integrals of the Ising model are not only "Derived from Geometry" (globally nilpotent), but actually correspond to "Special Geometry" (homomorphic to their formal adjoint). This raises the question of seeing if these "special geometry" Ising-operators, are "special" ones, reducing, in fact systematically, to (selected, k-balanced, ...) $_{q+1}F_q$ hypergeometric functions, or correspond to the more general solutions of Calabi-Yau equations.Comment: 35 page

Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment

We continue our study of the parabolic Anderson equation ¿u/¿t =k¿u+¿¿u for the space-time field u: Zd ×[0,8) ¿ R, where k ¿ [0,8) is the diffusion constant, ¿ is the discrete Laplacian, ¿ ¿ (0,8) is the coupling constant, and ¿ : Zd ×[0,8)¿R is a space-time random environment that drives the equation. The solution of this equation describes the evolution of a "reactant" u under the influence of a "catalyst" ¿, both living on Zd. In earlier work we considered three choices for ¿: independent simple random walks, the symmetric exclusion process, and the symmetric voter model, all in equilibrium at a given density. We analyzed the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of u w.r.t. ¿ , and showed that these exponents display an interesting dependence on the diffusion constant k, with qualitatively different behavior in different dimensions d. In the present paper we focus on the quenched Lyapunov exponent, i.e., the exponential growth rate of u conditional on ¿ . We first prove existence and derive some qualitative properties of the quenched Lyapunov exponent for a general ¿ that is stationary and ergodic w.r.t. translations in Zd and satisfies certain noisiness conditions. After that we focus on the three particular choices for ¿ mentioned above and derive some more detailed properties.We close by formulating a number of open problems