Q-matrices and transition matrices

Definition 2.1.1 A Q-matrix on a finite or countable state space I is a real-valued matrix (qij, i, j ∈ I) with:

non-positive diagonal entries qii ≤ 0, i ∈ I,

non-negative off-diagonal entries qij ≥ 0, i ≠ j, i, j ∈ I,

the row zero-sum condition: −qii = Σj∈I:j≠iqij, i.e. Σjqij = 0 for all i ∈ I.

For i ≠ j, the value qij represents the jump, or transition rate from state i to j. The value −qii = Σj:j≠iqij is denoted by qi (we will see that it represents the total jump, or exit rate from state i). A Q-matrix will be denoted by Q (a common abuse of notation). As in Chapter 1, we will denote by I the unit matrix.

In a general theory of countable continuous-time Markov chains, the row zerosum condition Σj∈I:j≠iqij = −qii presumes that the series Σj:j≠iqij < ∞. However, a substantial part of the theory can be developed when the equality in this condition is relaxed to the upper bound Σjqij ≤ 0, i.e. qi ≥ Σj:j≠iqij for all i ∈ I. Then a Q-matrix satisfying the row zero-sum condition is called conservative; we will omit this term in the present volume, as we will not consider non-conservative Q-matrices.

The Markov property and its immediate consequences

Typically, the subject of Markov chains represents a logical continuation from a basic course of probability. We will study a class of random processes describing a wide variety of systems of theoretical and practical interest (and sometimes simply amusing). The fact that deep insight into the subject is possible without using sophisticated mathematical tools may also be an explanation of why Markov chains are popular in so many different disciplines which are seemingly remote from pure mathematics.

The basic model for the first half of the book will be a system which changes state in discrete time, according to some random mechanism. The collection of states is called a state space and throughout the whole book will be assumed finite or countable; we will denote it by I. Each i ∈ I is called a state; our system will always be in one of these states. Sometimes we will know what state the system occupies and sometimes only that the system is in state i with some probability.

Homogenization is a theory about approximating solutions of a differential equation with rapidly varying coefficients by a solution of a constant coefficient differential equation of a similar nature. The simplest example of its kind is the solution u ϵ of the equation on [0, ∞] ⨯ ℝ. The function a(·) is assumed to be uniformly positive, continuous and periodic of period 1.

1. Introduction

The Vlasov equation. The evolution of N identical particles in phase space with coordinates .(qi ; pi)

i = 1, 2,....,N, may be described by an evolution equation for their joint probability distribution function. Integrating over all but one of the particle phase-space coordinates yields an evolution equation for the single-particle probability distribution function (PDF). This is the Vlasov equation.

The solutions of the Vlasov equation reflect its heritage in particle dynamics, which may be reclaimed by writing its many-particle PDF as a product of delta functions in phase space. Any number of these delta functions may be integrated out until all that remains is the dynamics of a single particle in the collective field of the others. In plasma physics, this collective field generates the total electromagnetic properties and the self-consistent equations obeyed by the single particle PDF are the Vlasov–Maxwell equations.

1. Introduction Zabusky and Kruskal [1965] introduced the concept of a soliton—a spatially localized solution of a nonlinear partial differential equation with the property that this solution always regains its initial shape and velocity after interacting with another localized disturbance. They were led to the concept of a soliton by their careful computational study of solutions of the Korteweg–de Vries (or KdV) equation, (See [Zabusky 2005] for his summary of this history.) After that breakthrough, they and their colleagues found that the KdV equation has many remarkable properties, including the property discovered by Gardner, Greene, Kruskal and Miura [Gardner et al. 1967]: the KdV equation can be solved exactly, as an initial-value problem, starting with arbitrary initial data in a suitable space. This discovery was revolutionary, and it drew the interest of many people. We note especially the work of Zakharov and Faddeev [1971], who showed that the KdV equation is a nontrivial example of an infinite-dimensional Hamiltonian system that is completely integrable. This means that under a canonical change of variables, the original problem can be written in terms of action-angle variables, in which the action variables are constants of the motion, while the angle variables evolve according to nearly trivial ordinary differential equations (ODEs).

We discuss the relevance of the classical Riccati substitution to the spectral edge statistics in some fundamental models of one-dimensional random Schrödinger and random matrix theory.

1. Introduction

The Riccati map amounts to the observation that the Schr¨odinger eigenvalue problemis transformed into the first order relation

upon setting. That this simple fact has deep consequences for the problem of characterizing the spectrum of Q with a random potential q has been known for some time. It also turns out to be important for related efforts in random matrix theory (RMT). We will describe some of the recent progress on both fronts.

Random operators of type Q arise in the description of disordered systems. Their use goes back to Schmidt [1957], Lax and Phillips [1958], and Frisch and Lloyd [1960] in connection with disordered crystals, represented by potentials in the form of trains of signed random masses, randomly placed on the line. Consider instead the case of white noise potential, q(x) =b′(x) with a standard brownian motion x ⟼ b(x), which may be viewed as a simplifying caricature of the above.

1. Introduction

Everyone is familiar with turbulence in one form or another. Airplane passengers encounter it in wintertime as the plane begins to shake and is jerked in various directions. Thermal currents and gravity waves in the atmosphere create turbulence encountered by low-flying aircraft. Turbulent drag also prevents the design of more fuel-efficient cars and aircrafts. Turbulence plays a role in the heat transfer in nuclear reactors, causes drag in oil pipelines and influence the circulation in the oceans as well as the weather.

In daily life we encounter countless other examples of turbulence. Surfers use it to propel them and their boards to greater velocities as the wave breaks and becomes turbulent behind them and they glide at great speeds down the unbroken face of the wave. This same wave turbulence shapes our beaches and carries enormous amount of sand from the beach in a single storm, sometime to dump it all into the nearest harbor. Turbulence is harnessed in combustion engines in cars and jet engines for effective combustion and reduced emission of pollutants.

1. Introduction

The equation of Camassa and Holm (CH) [1993; 1994] is an approximate one-dimensional description of unidirectional propagation of long waves in shallow water.

Introduction

Many distinguished authors have made notable contributions to the stochastic Burgers equation, of which a small sample appears in our very short bibliography. It is not our purpose to review those contributions; it is perhaps appropriate that we underline here that which seems to us the novelty of our approach.

We start from the viewpoint of geometrization of inertial evolution initiated in [Arnold 1966] and systematically developed in [Ebin and Marsden 1970; Brenier 2003; Constantin and Kolev 2002], based on infinite-dimensional Riemannian geometry; the classical approach of [Ebin and Marsden 1970] is to use Banach-modeled manifold theory; inherent difficulties appear in the construction of exponential charts and in the introduction of appropriate function spaces. We circumvent these difficulties by using the viewpoint [Malliavin 2007] of Itô charts, Itô atlas; in short Itô calculus makes it possible to compute any derivative of a smooth function f on the path p of a diffusion from the unique knowledge of its restriction f|p. Then no more function spaces are a priori introduced: the path of the diffusion constructs dynamically its canonical tangent space, built from the evolution of the system.

This volume is dedicated to Henry McKean, on the occasion of his seventyfifth birthday. His wide spectrum of interests within mathematics is reflected in the variety of theory and applications in these papers, discussed in the Tribute on page xv. Here we comment briefly on the papers that make up this volume, grouping them by topic. (The papers appear in the book alphabetically by first author.)

Since the early 1970s, the subject of completely integrable systems has grown beyond all expectations. The discovery that the Kortweg – de Vries equation, which governs shallow-water waves, has a complete system of integrals of motion has given rise to a search for other such evolution equations. Two of the papers in this volume, one by Boutet de Monvel and Shepelsky and the other by Loubet, deal with the completely integrable system discovered by Camassa and Holm. This equation provides a model describing the shallow-water approximation in inviscid hydrodynamics. The unknown function u.(x; t) refers to the horizontal fluid velocity along the x-direction at time t . The first authors show that the solution of the CH equation in the case of no breaking waves can be expressed in parametric form in terms of the solution of an associated Riemann– Hilbert problem. This analysis allows one to conclude that each solution within this class develops asymptotically into a train of solitons.

Loubet provides a technical tour de force, extending previous results of McKean on the Camassa–Holm equation. More specifically, he gives an explicit formula for the velocity profile in terms of its initial value, when the dynamics are defined by a Hamiltonian that is the sum of the squares of the reciprocals of a pair of eigenvalues of an associated acoustic equation.

1. Introduction

In the theory of critical phenomena it is usually assumed that a physical system near a continuous phase transition is characterized by a single length scale (the correlation length) in terms of which all other lengths should be measured. When combined with the experimental observation that the correlation length diverges at the phase transition, this simple but strong assumption, known as the scaling hypothesis, leads to the belief that at criticality the system has no characteristic length, and is therefore invariant under scale transformations. This suggests that all thermodynamic functions at criticality are homogeneous functions, and predicts the appearance of power laws.

It also implies that if one rescales appropriately a critical lattice model, shrinking the lattice spacing to zero, it should be possible to obtain a continuum model, known as the scaling limit. The scaling limit is not restricted to a lattice and may possess more symmetries than the original model. Indeed, the scaling limits of many critical lattice models are believed to be conformally invariant and to correspond to Conformal Field Theories (CFTs).

1. Introduction

The most powerful method for constructing explicit periodic and quasiperiodic solutions of soliton equations is based on the finite-gap or algebro-geometric approach, developed by Novikov [1974], Dubrovin et al. [1976b], Its and Matveev [1975], Lax [1975], and McKean and van Moerbeke [1975] for 1+1 systems, and extended by Krichever in 1976 for 2+1 systems like KP. Already in 1976 new ideas were formulated on how to extend this approach to the 2 + 1 systems associated with the spectral theory of the 2D Schr¨odinger operator restricted to one energy level; see [Manakov 1976; Dubrovin et al. 1976a]. These ideas were developed in 1980s by several people in Moscow's Novikov Seminar, as discussed see below. The “spectral data” characterizing the associated Lax-type operators consist of a Riemann surface (spectral curve) equipped with a selected set of points (divisor of poles, infinities). In the finite gap case this Riemann surface has finite genus, and the number of selected point is also finite.

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).