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El. knyga: Lace Expansion and its Applications: Ecole d'Ete de Probabilites de Saint-Flour XXXIV - 2004

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Three series of lectures were given at the 34th Probability Summer School in Saint-Flour (July 624, 2004), by the Professors Cerf, Lyons and Slade. We have decided to publish these courses separately. This volume contains the course of Professor Slade. We cordially thank the author for his performance at the summer school, and for the redaction of these notes. 69 participants have attended this school. 35 of them have given a short lecture. The lists of participants and of short lectures are enclosed at the end of the volume. The Saint-Flour Probability Summer School was founded in 1971. Here are the references of Springer volumes which have been published prior to this one. All numbers refer to theLecture Notes in Mathematics series, except S-50 which refers to volume 50 of the Lecture Notes in Statistics series. 1971: vol 307 1980: vol 929 1990: vol 1527 1998: vol 1738 1973: vol 390 1981: vol 976 1991: vol 1541 1999: vol 1781 1974: vol 480 1982: vol 1097 1992: vol 1581 2000: vol 1816 1975: vol 539 1983: vol 1117 1993: vol 1608 2001: vol 1837 & 1851 1976: vol 598 1984: vol 1180 1994: vol 1648 2002: vol 1840 1977: vol 678 1985/86/87: vol 1362 & S-50 1995: vol 1690 2003: vol 1869 1978: vol 774 1988: vol 1427 1996: vol 1665 2004: vol.

Recenzijos

From the reviews:









"The book consists mainly of materials for an advanced master course. The book focusses on the modeling of mechanics, where the concept of a model shall be understood in its broadest sense, namely as a mathematical structure which describes mechanical phenomena. A list of notations and an index are helpful to the reader. The little book with about 200 pages tackles many theories and concepts, without going much into detail." (Albrecht Bertram, Zentralblatt MATH, Vol. 1115 (17), 2007)



"This work is based on the authors lectures on the lace expansion and its applications . It contains the recent developments in the analysis of the lace expansion and the scaling limits of the critical objects discovered since the publication . It also contains further applications of the lace expansion to other models and an extensive list of references. The subject is still growing, and studying the lecture notes by Slade is a good starting point to learn the subject." (Akira Sakai, Mathematical Reviews, Issue 2007 m)

Simple Random Walk
1(6)
Asymptotic Behaviour
1(4)
Universality and Spread-Out Models
5(2)
The Self-Avoiding Walk
7(12)
Asymptotic Behaviour
7(7)
Differential Inequalities and the Bubble Condition
14(5)
The Lace Expansion for the Self-Avoiding Walk
19(12)
Inclusion-Exclusion
19(2)
Expansion
21(3)
Laces and Resummation
24(3)
Transformations
27(4)
Diagrammatic Estimates for the Self-Avoiding Walk
31(10)
The Diagrammatic Estimates
31(2)
Proof of the Diagrammatic Estimates
33(8)
Convergence for the Self-Avoiding Walk
41(16)
Random-Walk Estimates
41(5)
Convergence of the Expansion
46(8)
Finite Bubble vs Small Bubble
54(1)
Differential Equality and the Bubble Condition
54(3)
Further Results for the Self-Avoiding Walk
57(10)
The Self-Avoiding Walk in Dimensions d > 4
57(4)
A Self-Avoiding Walk in Dimension d = 1
61(1)
Nearest-Neighbour Attraction
62(1)
Networks of Self-Avoiding Walks
63(4)
Lattice Trees
67(10)
Asymptotic Behaviour
67(4)
Differential Inequalities and the Square Condition
71(6)
The Lace Expansion for Lattice Trees
77(10)
Expansion for Lattice Trees
77(4)
Expansion for Lattice Animals
81(2)
Diagrammatic Estimates for Lattice Trees
83(4)
Percolation
87(22)
The Phase Transition
87(4)
Differential Inequalities
91(4)
Differential Inequalities and the Triangle Condition
95(4)
Proofs of the Differential Inequalities
99(10)
The Expansion for Percolation
109(16)
The Expansion
109(7)
The Diagrams
116(5)
Diagrammatic Estimates for Percolation
121(4)
Results for Percolation
125(16)
Critical Exponents
125(4)
The Critical Value
129(1)
The Incipient Infinite Cluster
130(3)
Percolation on Finite Graphs
133(8)
Oriented Percolation
141(10)
The Phase Transition
141(2)
The Infrared Bound and the Triangle Condition
143(3)
The Critical Value
146(1)
The Incipient Infinite Cluster
147(4)
Expansions for Oriented Percolation
151(10)
Inclusion-Exclusion
152(2)
Laces and Resummation
154(2)
Application of the Percolation Expansion
156(1)
The Upper Critical Dimension
157(1)
Diagrams for Oriented Percolation
158(3)
The Contact Process
161(10)
The Phase Transition
161(2)
Approximation by Oriented Percolation
163(1)
The Infrared Bound and the Triangle Condition
164(3)
Expansion for the Contact Process
167(2)
Diagrams for the Contact Process
169(2)
Branching Random Walk
171(12)
A Mean-Field Model
171(1)
Branching Random Walk
172(8)
Weakly Self-Avoiding Lattice Trees
180(3)
Integrated Super-Brownian Excursion
183(18)
Moment Measures of Branching Random Walk
183(2)
Critical Exponents and Generating Functions
185(4)
Construction of ISE
189(5)
Lattice trees and ISE
194(3)
Critical Percolation and ISE
197(4)
Super-Brownian Motion
201(10)
The Canonical Measure of SBM
201(2)
Moment Measures of the Canonical Measure
203(4)
Critical Oriented Percolation and SBM
207(2)
The Critical Contact Process and SBM
209(1)
Lattice Trees and SBM
210(1)
References 211(10)
Index 221


Gordon Slade is Professor at the University of British Columbia since 1999. Before he was Lecturer at the University of Virginia from 1985 to 1986 and  Professor at the McMaster University from 1986 to 1999. The Author has been awarded the UBC Killam Research Prize (Senior Science Category) in 2004 and the Prix de l'Institut Henri Poincaré--with Remco van der Hofstad--in2003. In 2003 he was Stieltjes Visiting Professor, in 1995 Coxeter-James Lecturer for the Canadian Mathematical Society. Since 2000 he is Fellow of the Royal Society of Canada. 
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