This is material for a course in Buzios, Brazil, Summer 2010, as part of:
Clay Mathematics Institute Summer School July 11 to August 7, 2010, and
XIV Escola Brasileira de Probabilidade August 2-7, 2010.
The course will consist of a 90-minute lecture each morning August 2-7 and a tutorial each afternoon
August 2-6.
Tutorials will be led by Roland Bauerschmidt and Jesse Goodman, University of
British Columbia.
There will be a guest lecture by Hugo Duminil-Copin (University of
Geneva) during the tutorial session on Tuesday, August 3rd.
Self-Avoiding Walks. The course will focus on rigorous results for self-avoiding walks on the d-dimensional integer lattice. The model is defined by assigning equal probability to all random walk paths starting from the origin and without self-intersections. These probability measures are not consistent as the path length is varied, and thus do not define a stochastic process; the model is combinatorial in nature. Despite its simple definition, the self-avoiding walk is difficult to study in a mathematically rigorous manner, and many of the important problems remain unsolved; they encompass many of the features and challenges of critical phenomena.
General references:
N. Madras and G. Slade,
The Self-Avoiding Walk, Birkhäuser, Boston, (1993).
G. Slade. The Lace Expansion and its Applications, Lecture Notes in Mathematics #1879. Springer, Berlin, (2006).
PDF file
B.D. Hughes,
Random walks and random environments, Volume 1, Oxford University Press, Oxford (1995).
Lecture 1.
Introduction to the self-avoiding walk model, connective constant, critical exponents,
dependence of critical behaviour on dimension, overview of what is known and of open problems.
References:
G. Slade, The Self-Avoiding Walk: A Brief Survey.
To appear in Surveys in Stochastic Processes,
Proceedings of the 33rd SPA Conference in Berlin, 2009, to be
published in the EMS Series of Congress Reports, eds. J. Blath, P. Imkeller,
S. Roelly. PDF file
N. Madras and G. Slade, The Self-Avoiding Walk, Chapter 1.
You can read it at google books.
Tutorial 1. Exercise sheet PDF file.
Lecture 2.
The Hammersley-Welsh upper bound on the number of n-step self-avoiding walks.
Although very far from the predicted power law behaviour, this 1962
bound has not been improved (for dimension d=2) in almost 50 years. Its elegant proof
employs the useful notion of bridges. These apply to the study of self-avoiding polygons.
References:
1) N. Madras and G. Slade, The Self-Avoiding Walk.
Sections 3.1-3.2. You can read it at google books.
2) J.M. Hammersley and D.J.A. Welsh, Further results on the rate of convergence
to the connective constant of the hypercubical lattice. Quart. J. Math. Oxford,
(2), 13, 108-110, (1962).
Tutorial 2: Guest lecture by Hugo Duminil-Copin on the following paper:
H. Duminil-Copin and S. Smirnov,
The connective constant of the honeycomb lattice equals
√ 2 +
√ 2
. Preprint (2010).
PDF file
Lectures 3 and 4.
Derivation and application of the lace expansion for self-avoiding walks.
The lace expansion has been used to prove that self-avoiding walks in dimensions
5 and higher behave like simple random walks. The mean-square displacement is
linear in the number of steps and the scaling limit is Brownian motion.
Reference:
G. Slade.
The Lace Expansion and its Applications, Lecture Notes in Mathematics #1879. Springer, Berlin, (2006).
Section 2.2 and Chapters 3,4,5.
PDF file
Tutorial 3. Exercise sheet PDF file. Tutorial 4. Exercise sheet PDF file.
Lecture 5.
Integral representations for the self-avoiding walk.
The two-point function for strictly self-avoiding walk or weakly self-avoiding walk
can be written as a Gaussian integral involving anti-commuting variables.
Such representations have been useful as the point of departure for renormalisation
group analysis.
Reference:
D.C. Brydges, J.Z. Imbrie, G. Slade. Functional integral representations for self-avoiding walk. Probability Surveys,
6, 34-61, (2009). PDF file
Tutorial 5. Exercise sheet PDF file.
Lecture 6.
Renormalisation group analysis.
Using the integral representation, renormalisation group methods have been
developed and used to prove inverse square decay for the critical two-point function of
the continuous-time weakly self-avoiding walk in dimension 4.
Reference:
D. Brydges and G. Slade. Renormalisation group analysis of weakly self-avoiding walk in dimensions four and higher. To appear in the Proceedings of the International Congress
of Mathematicians, Hyderabad, 2010. PDF file