There are still algebraic surprises lying concealed within the Toda flow that we have not yet described. A simple argument with matrix factorizations gives rise to new inverse variables, the Z-coordinates, together with new commuting vector fields. In these new variables, the Toda flows become explicit, straight line motions in $R^N$ for an appropriate dimension N (Toda flows on full matrices are also considered). Moreover, orbit limits, such as diagonal matrices, lie beyond the scope of standard variables (see, e.g., M below for Jacobi matrices), and the asymptotic analysis must proceed using ad hoc methods. Such orbit limits, however, belong to the domain of the new variables, and asymptotic computations are easily performed through local theory. As described in Remark 4.14, the methods in this section extend the purview of the Toda system substantially.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

The goal of this chapter is to give an elementary introduction to Hamiltonian mechanics, and particularly integrable Hamiltonian systems, with a view to describing various results that we need in analyzing the Toda algorithm. The reader is encouraged to consult references such as Abraham and Marsden (1978), Arnold (1978), Kirillov (2004), Moser and Zehnder (2005) and Warner (1983) for a more detailed presentation.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

We now turn to studying the Toda lattice. As noted in Chapter 1, the (open) Toda lattice was introduced by H. Toda in 1967 and describes the motion of N particles ${(x_i)}_{i=1}^N$ on the real line, generated by the Hamiltonian $H_{\text{Toda}}(x,y)=\frac{1}{2}{\displaystyle \sum _{i=1}^N{y}_i^2}+{\displaystyle \sum _{i=1}^{N-1}{e}^{x_i-x_{i+1}}}$ on the symplectic manifold $M^{(2N)}=\left({ℝ}^{2N},={\displaystyle \sum _{i=1}^N{x}_i}\wedge y_i\right)$.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

We now introduce the results from random matrix theory that are needed to prove Theorem 6.2 and Proposition 6.5 in the next chapter.Let H be an N x N Hermitian (or real symmetric) matrix with eigenvalues ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_N$ and let ${\beta }_1,{\beta }_2,\dots ,{\beta }_N$ denote the absolute value of the first components of the normalized eigenvectors.We assume the entries of H are distributed according to an invariant or generalized Wigner ensemble (see Section 5.1).