Semiclassical analysis, Witten laplacians, and statistical mechanics

Bernard Helffer9812380981, 9789812380982, 9789812776891

This important book explains how the technique of Witten Laplacians may be useful in statistical mechanics. It considers the problem of analyzing the decay of correlations, after presenting its origin in statistical mechanics. In addition, it compares the Witten Laplacian approach with other techniques, such as the transfer matrix approach and its semiclassical analysis. The author concludes by providing a complete proof of the uniform Log–Sobolev inequality.

---

Table of contents :
Contents……Page 8
Preface……Page 6
1.1 Laplace integrals……Page 11
1.2 The problems in statistical mechanics……Page 13
1.3 Semi-classical analysis and transfer operators……Page 16
1.4 About the contents……Page 17
2.1 De Rham Complex……Page 19
2.2 Witten Complex……Page 23
2.3 Witten Laplacians……Page 24
2.4 Semi-classical considerations……Page 25
2.5 An alternative point of view : Dirichlet forms……Page 26
2.6 A nice formula for the covariance……Page 27
2.7 Notes……Page 31
3.1 The Curie-Weiss model……Page 33
3.2 The 1-d Ising model……Page 35
3.3 The 2-d Ising model……Page 38
3.4 Notes……Page 39
4.2.1 Standard results……Page 41
4.2.2 Transition between the convex case and the double well case……Page 43
4.3 The method of transfer operators……Page 44
4.4 Elementary properties of operators with integral kernels……Page 45
4.5 Elementary properties of the transfer operator……Page 49
4.6 Operators with strictly positive kernel and application……Page 50
4.7 Thermodynamic limit……Page 52
4.8 Mean value……Page 53
4.9 Pair correlation……Page 54
4.10 2-dimensional lattices……Page 55
4.11 Notes……Page 59
5.2 Explicit computations for the harmonic Kac operator……Page 61
5.3 Harmonic approximation for the transfer operator……Page 65
5.4 WKB constructions for the transfer operator……Page 69
5.5 The case of the Schrodinger operator in dimension 1……Page 73
5.6 Harmonic approximation for the transfer operator:upper bounds……Page 75
5.7 First conclusions about the splitting……Page 77
5.8 Some elements about the decay……Page 78
5.9.2 Comparison of various problems……Page 80
5.9.3 Upper bound of the splitting……Page 81
5.10 Notes……Page 83
6.2 Selfadjoint operators spectrum and spectral decomposition……Page 87
6.3 Discrete spectrum essential spectrum……Page 93
6.4 Essentially selfadjoint operators……Page 96
6.5.1 The free Laplacian……Page 98
6.5.2 The harmonic oscillator……Page 99
6.6 More on selfadjointness……Page 101
6.7 The max-min principle……Page 103
6.8 Compactness……Page 104
6.9 Notes……Page 107
7.1 Introduction……Page 111
7.2 Log-Sobolev inequalities in the strictly convex case……Page 113
7.3 Around Herbst’s argument : necessary conditions for log-Sobolev inequalities……Page 125
7.4 Extension of the Bakry-Emery argument : convexity at infinity……Page 128
7.5 The case of the circle……Page 130
7.6 The case of the line……Page 133
7.7 General remarks……Page 138
7.8 Notes……Page 140
8.1 Introduction……Page 143
8.2 Lower bound for the spectrum of the Witten Laplacian……Page 146
8.3 Uniform estimates for a family of 1-dimensional Witten Laplacians……Page 149
8.4 A proof of the decay of correlations……Page 152
8.5 Generalized Brascamp-Lieb inequality……Page 157
8.6 Notes……Page 158
9.1 Introduction and preliminaries……Page 161
9.2 Some log-Sobolev inequality for effective single spin phase……Page 162
9.3 The role of the decay estimates for log-Sobolev inequality……Page 165
9.4 Second part of the proof of the log-Sobolev inequality……Page 168
9.5 Conclusion……Page 176
9.6 Notes……Page 177
Bibliography……Page 179
Index……Page 187