Henry McKean was one of the earliest contributors to the field of “integrable PDEs”, whose origin for simplicity we shall place in the late 1960s. One way in which Henry conveyed the stunning and powerful discovery of a linearizing change of variables was by choosing Isaiah 40:3-4 as an epigram for [McKean 1979]: The voice of him that crieth in the wilderness, Prepare ye the way of the Lord, make straight in the desert a highway for our God. Every valley shall be exalted and every mountain and hill shall be made low: and the crooked shall be made straight and the rough places plain. Thus, on this contribution to a volume intended to celebrate Henry's many fundamental achievements on the occasion of his birthday, my title. I use the word line in the extended sense of “linear flow”, of course, since no projective line can be contained in an abelian variety—the actual line resides in the universal cover.

1. Background and preliminaries

The study of the unitary ensembles (UE) of random matrices [Mehta 1991], begins with a family of probability measures on the space of N × N Hermitian matrices. The measures are of the form where the function Vt is a scalar function, referred to as the potential of the external field, or simply the “external field” for short. Typically it is taken to be a polynomial, and written as follows. where the parameters are assumed to be such that the integral converges. For example, one may suppose thatis even, and tυ > 0. The partition function , is the normalization factor which makes the UE measures be probability measures. Expectations of conjugation invariant matrix random variables with respect to these measures can be reduced, via the Weyl integration formula, to an integration against a symmetric density over the eigenvalues which has a form proportional to (1-1).

ABSTRACT. Given a constrained minimization problem, under what conditions does there exist a related, unconstrained problem having the same minimum points? This basic question in global optimization motivates this paper, which answers it from the viewpoint of statistical mechanics. In this context, it reduces to the fundamental question of the equivalence and nonequivalence of ensembles, which is analyzed using the theory of large deviations and the theory of convex functions.

In a 2000 paper appearing in the Journal of Statistical Physics,we gave necessary and sufficient conditions for ensemble equivalence and nonequivalence in terms of support and concavity properties of the microcanonical entropy. In later research we significantly extended those results by introducing a class of Gaussian ensembles, which are obtained from the canonical ensemble by adding an exponential factor involving a quadratic function of the Hamiltonian. The present paper is an overview of our work on this topic. Our most important discovery is that even when the microcanonical and canonical ensembles are not equivalent, one can often find a Gaussian ensemble that satisfies a strong form of equivalence with the microcanonical ensemble known as universal equivalence. When translated back into optimization theory, this implies that an unconstrained minimization problem involving a Lagrange multiplier and a quadratic penalty function has the same minimum points as the original constrained problem.

The results on ensemble equivalence discussed in this paper are illustrated in the context of the Curie–Weiss–Potts lattice-spin model.

1. Introduction

Oscar Lanford, at the beginning of his groundbreaking paper [Lanford 1973], describes the underlying program of statistical mechanics.

1. Introduction

In this paper we shall present a brief survey of a number of results on direct and inverse problems for canonical integral and differential systems that have been obtained by the authors over the past several years. We shall not attempt to survey the literature, which is vast, or to compare the methods surveyed here with other approaches. The references in [Arov and Dym 2004; 2005b; 2005c] (the last of which is a survey article) may serve at least as a starting point for those who wish to explore the literature.

where H(t) is an m x m locally summable mvf (matrix valued function) that is positive semidefinite a.e. on the interval [0, d], j is an m x m signature matrix, i.e., J * J and J * J = Im, and y(t, ƛ) is a k x m mvf.

We present a Riemann-Hilbert problem formalism for the initial value problem for the Camassa-Holm equation ut-utxx + 2ωux + 3uux= 2uxuxx + uuxxx on the line (CH), where ω is a positive parameter. We show that, for all ω > 0, the solution of this initial value problem can be obtained in a parametric form from the solution of some associated Riemann-Hilbert problem; that for large time, it develops into a train of smooth solitons; and that for small ω, this soliton train is close to a train of peakons, which are piecewise smooth solutions of the CH equation for ω = 0

1. Introduction

The main purpose of this paper is to develop an inverse scattering approach, based on an appropriate Riemann–Hilbert problem formulation, for the initial value problem for the Camassa–Holm (CH) equation [Camassa and Holm 1993] on the line, whose form is where ω is a positive parameter. The CH equation is a model equation describing the shallow-water approximation in inviscid hydrodynamics. In this equation u = u(x, t) is a real-valued function that refers to the horizontal fluid velocity along the x direction (or equivalently, the height of the water's free surface above a flat bottom) as measured at time t . The constant ω is related to the critical shallow water wave speed, where g is the acceleration of gravity and h0 is the undisturbed water depth; hence, the case ω >0 is physically more relevant than the case ω = 0, though the latter has attracted more attention in the mathematical studies due to interesting specific features such as the existence of peaked (nonanalytic) solitons.

Consider n nonintersecting Brownian motions on ℝ, which leave from p definite points and are forced to end up at q points at time t = 1. When n → ∞, the equilibrium measure for these Brownian particles has its support on p intervals, for t ∽ 0, and on q intervals, for t ∽ 1. Hence it is clear that, when t evolves, intervals must merge, must disappear and be created, leading to various phase transitions between times t = 0 and 1.

Near these moments of phase transitions, there appears an infinite-dimensional diffusion, a Markov cloud, in the limit n ↗ ∞, which one expects to depend only on the nature of the singularity associated with this phase change. The transition probabilities for these Markov clouds satisfy nonlinear PDE’s, which are obtained from taking limits of the Brownian motion model with finite particles; the finite model is closely related to Hermitian matrix integrals, which themselves satisfy nonlinear PDE’s. The latter are obtained from investigating the connection between the Karlin-McGregor formula, moment matrices, the theory of orthogonal polynomials and the associated integrable systems. Various special cases are provided to illustrate these general ideas. This is based on work by Adler and van Moerbeke.

1. Introduction

This lecture in honor of Henry McKean forms a step in the direction of understanding the behavior of nonintersecting Brownian motions on ℝ (Dyson's Brownian motions), when the number of particles tends to ∞. It explains a novel interface between diffusion theory, integrable systems and the theory of orthogonal polynomials. These subjects have been at the center of Henry McKean's oeuvre.

1. In the beginning there was Gauss

In the year 1985, one of us had the luxury of attending a graduate course on Elliptic Functions given by Henry McKean at the Courant Institute. Among the many beautiful results he described in his unique style, there was a calculation of Gauss: take two positive real numbers a and b, with a > b, and form a new pair by replacing a with the arithmetic mean . and b with the geometric mean. Then iterate:

starting with a = a and b0 = b. Gauss [1799] was interested in the initial conditions a = 1 and . The iteration generates a sequence of algebraic numbers which rapidly become impossible to describe explicitly; for instance is a root of the polynomial The numerical behavior is surprising; a6 and b6 agree to 87 digits. It is simple to check that

and then he recognized the reciprocal of this number as a numerical approximation to the elliptic integral

It is unclear to the authors how Gauss recognized this number: he simply knew it. (Stirling's tables may have been a help; [Borwein and Bailey 2003] contains a reproduction of the original notes and comments.) He was particularly interested in the evaluation of this definite integral as it provides the length of a lemniscate. In his diary Gauss remarked, ‘This will surely open up a whole new field of analysis’ [Cox 1984; Borwein and Borwein 1987].

1. Introduction

Consider a birth and death process, i.e., a discrete time Markov chain on the nonnegative integers, with a one step transition probability matrix ℙ. There is then a time-honored way of writing down the n-step transition probability matrix ℙn in terms of the orthogonal polynomials associated to ℙ and the spectral measure. This goes back to Karlin and McGregor [1957] and, as they observe, it is nothing but an application of the spectral theorem. One can find some precursors of these powerful ideas, see for instance [Harris 1952; Ledermann and Reuter 1954]. Inasmuch as this is such a deep and general result, it holds in many setups, such as a nearest neighbours random walk on the Nth-roots of unity. In general this representation of ℙn allows one to relate properties of the Markov chain, such as recurrence or other limiting behaviour, to properties of the orthogonality measure.

In the few cases when one can get one's hands on the orthogonality measure and the polynomials themselves this gives fairly explicit answers to various questions.

The two main drawbacks to the applicability of this representation (to be recalled below) are:

• a) typically one cannot get either the polynomials or the measure explicitly.

• b) the method is restricted to “nearest neighbour” transition probability chains that give rise to tridiagonal matrices and thus to orthogonal polynomials.