DISORDERED SYSTEMS, RANDOM SPATIAL PROCESSES AND SOME APPLICATIONS

The main topic of these lectures is the continuum scaling limit of planar lattice models. One reason why this topic occupies an important place in the theory of probability and mathematical statistical physics is that scaling limits provide the link between statistical mechanics and (Euclidean) field theory. In order to explain the main ideas behind the concept of scaling limit, I will focus on a "toy" model that exhibits the typical behavior of statistical mechanical models at and near the critical point. This "toy" model, known as the random walk loop soup, turns out to be interesting in its own right. It can be described as a Poissonian ensemble of lattice loops, or a lattice gas of loops since it fits within the ideal-gas framework of statistical mechanics. After introducing the model and discussing some interesting connections with the discrete Gaussian free field, I will present some results concerning its scaling limit, which leads to a Poissonian ensemble of continuum loops known as the Brownian loop soup. The latter was introduced by Lawler and Werner, and is a very interesting object with connections to the Schramm-Loewner Evolution and various models of statistical mechanics. In the second part of the lectures, I will use the Brownian loop soup to construct a family of functions that behave like correlation functions of a conformal field. I will then use these functions and their derivation to introduce the concept of conformal field and to explore the connection between scaling limits and conformal fields.

I will consider stochastic evolutions of particles that are confined in a bounded region by means of different mechanisms at the boundary. The motivation comes from one of the basic questions in statistical mechanics concerning the structure of states which are stationary but not in thermal equilibrium. These states are characterized by the presence of a steady current flowing through the system and the Fouries (or Fick) law is expected to hold. I will present different ways to impose a current and then I will discuss and analyze the different effects so produced both at the microscopic and macroscopic level. Rather surprisingly, these studies have generated many interesting problems both mathematically and conceptually. There are many papers on this subject and I will focus on some of them.

In this course, we will review recent results in the theory of geometric representations of low-dimensional spin systems such as Potts, spin O(n) and Ising models. We will introduce random-cluster models, loop O(n)-models and random current representations and will explain how these models can be used to study the spin models at criticality and determine in particular whether their phase transition is continuous or not.

A self-contained review will be given about the equilibrium and off equilibrium properties of statistical mechanics systems, both in the case of ferromagnetic and disordered ones. A systematic use of interpolation arguments and convexity will be the main tool. The case of mean field will be treated throughly. The possible extensions to short range interaction will be pointed out.

Stationary non-equilibrium states describe steady flows through macroscopic systems. Although they represent the simplest generalization of equilibrium states, they exhibit a variety of new phenomena. Within a statistical mechanics approach, these states have been the subject of several theoretical investigations, both analytic and numerical. The macroscopic fluctuation theory, based on a formula for the probability of joint space-time fluctuations of thermodynamic variables and currents, provides a unified macroscopic treatment of such states for driven diffusive systems. The course will give a review of this theory including its main predictions and some relevant applications.

The course consists of two parts. In the first one we give an introduction to the Renormalization Group as a method to study quantum field theory and statistical mechanics models at critical temperature. In the second part we apply these ideas to proving existence and uniqueness of solutions of stochastic PDE's driven by space time white noise. Examples are the KPZ and \phi^4_3 models.

In this minicourse we discuss some old results concerning the location of zeros of partition functions (or moment generating functions) in certain statistical mechanics models and their possible connections to the Riemann Hypothesis (RH). A standard reformulation of the RH is as follows. The (two-sided) Laplace transform of a certain specific function \Psi on the real line is automatically an entire function on the complex plane; the RH is equivalent to this transform having only pure imaginary zeros. Also \Psi is a positive integrable function, so (modulo a multiplicative constant C) is a probability density function. A (finite) Ising model (with pair ferromagnetic interactions) is a specific type of probability measure P on the points S=(S_1,...,S_N) with each S_j = +1 or -1. The Lee-Yang theorem implies that that for non-negative a_1, ..., a_N, the Laplace transform of the induced probability distribution of a_1 S_1 + ... + a_N S_N has only pure imaginary zeros. The big question here is whether it's possible to find a sequence of Ising models so that the limit as N tends to \infty of such distributions has density exactly C \Psi. The course will focus on questions of this sort. Here are some background references: C. Newman, Z. f. Wahrschein. (Prob. Th. Re. Fields) 33 (1975) 25-93 (see, esp. p. 90) C. Newman, Proc. Amer. Math. Soc. 61 (1976) 245-251 C. Newman, Constr. Approx. 7 (1991) 389-399 A. Odlyzko, Num. Algorithms 25 (2000) 293-303

I will describe recent results and works in progress on the absence/presence of phase transitions in systems with spatially non homogeneous interactions. In a first part I will consider the d ≥ 2 nearest neighbor ferromagnetic Ising model under the action of a non negative, space dependent magnetic field. It will be shown that at low temperatures the occurrence of a phase transition depends critically on the rate at which the magnetic field vanishes at infinity. In a second part I will consider two dimensional systems with ”very small, short range vertical interactions” while the horizontal interaction is described by a two body Kac potential. I will first study the Ising case and then extend the analysis to a continuous system where on each horizontal line there is a system of hard rods with attractive Kac pair interaction.The goal is to prove that such a system has the phase transition predicted by van der Waals, which at this moment is still under study.

Thermodynamics and statistical mechanics are universal, and we expect them to apply also to gravitational phenomena. But our understanding of gravity is given by general relativity, where temporal evolution is given in a form incompatible with standard thermodynamics and statistical mechanics. All principles of thermodynamics become wrong or meaningless in gravity: temperature fails to be constant at equilibrium (Tolman effect), energy is ill-defined, and it is not clear which among the many notions of time, if any, may underpin the second law. I discuss partial successes and open problems in the effort to extend the foundations of thermodynamics and statistical mechanics to a wider context encompassing relativistic gravity. Among the key ideas aiming to do so, is the idea that the physical flow of time of the second law is determined by, rather than determines, the statistical state of a covariant system. In the quantum field theoretical context, it is given by the Tomita flow generated by a state on the observables' algebra, and the origin of time's flow is related, via a theorem by Alain Connes, to the underlying quantum non-commutativity of the observable algebra.

In 1986, Kardar, Parisi and Zhang introduced a non-linear stochastic PDE to describe dynamics of interface motion. This KPZ equation has been studied intensively and extensively since then, but recently its one-dimensional version has been attracting particular attention because of its tractability and connections to various areas of mathematics and physics. In this lecture I will explain part of these developments . I will mainly focus on the height distribution for the KPZ equation. I will also explain the underlying algebraic structure such as random matrix theory, Schur process and Macdonald process in connection to related discrete interacting particle systems like ASEP and q-boson zero range process.

The course will cover:
-- determinantal processes and the statistical mechanics of line ensembles,
-- growth models and directed polymers in a random medium,
-- the Kardar-Parisi-Zhang stochastic PDE, sharp wedge initial conditions,
-- duality for interacting Brownian motions and other models in the KPZ class,
-- multi-component KPZ equations.

The aim of this lecture series is to introduce the subject of spin glasses, and more generally the statistical mechanics of quenched disorder, as a problem of general interest to physicists and mathematicians from multiple disciplines and backgrounds. Despite years of study, the physics and mathematics of quenched disorder remains poorly understood, and represents a major gap in our understanding of the condensed state of matter. While there are many active areas of investigation in this field, I will narrow the focus of this series to some aspects of our current level of understanding of the low-temperature equilibrium structure of realistic (i.e., finite-dimensional) spin glasses. I will begin with a brief review of the basic features of spin glasses and what is known experimentally. I will then turn to the problem of understanding the nature of the spin glass phase --- if it exists. The central question to be addressed is the nature of broken symmetry in these systems. Parisi's replica symmetry breaking approach, now mostly verified for mean field spin glasses, attracted great excitement and interest as a novel and exotic form of symmetry breaking. But does it hold also for real spin glasses in finite dimensions? This has been a subject of intense controversy, and although the issues surrounding it have become more sharply defined in recent years, it remains an open question. I will explore this problem, introducing new mathematical constructs such as the metastate along the way, as well as related questions such as the number of pure states and free energy fluctuations. If time permits, we will conclude with an examination of some of the applications of spin glass mathematics to problems in computer science, biology, and other fields.

Abstract: The Brownian web is the collection of one-dimensional coalescing Brownian motions starting from every point in space-time. Originally conceived by Arratia in the context of the one-dimensional voter model and its dual coalescing random walks, the Brownian web has since been shown to arise in the scaling limit of many one-dimensional interacting particle systems with coalescent interaction, including zero-temperature dynamics of Ising and Potts models, true self-avoiding random walks, drainage networks, Hastings-Levitov planar aggregation models, and super-critical oriented percolation. The Brownian net is an extension of the Brownian web, which also allows for branching of the Brownian motions. It has been shown to arise in the scaling limit of many one-dimensional interacting particle systems with branching-coalescing interactions, including the voter model with selection, dynamics of Ising and Potts models with boundary nucleation, and one-dimensional random walks in i.i.d. space-time random environments. The goal of the lecture series is to introduce the Brownian web and the Brownian net, discuss some of their properties, and study how they arise in the scaling limits of various models of interest.

This mini-course will review the use of the second moment method in proving limit laws for extremes of the two dimensional GFF; parallels and differences with branching random walks will be discussed.