Statistical physics, combinatorics and probability: from discrete to continuous models

Lectures

On this page: extended courses, mini-courses, seminars.

Thanks to our dedicated lecturers, much material (slides, notes, etc) is available from this page.

Extended courses

Philippe Di Francesco, Integrable Combinatorics. Wednesdays 9, 16, 23 and 30 September 2009, 10-12h and 14-16h. Lecture notes:
Philippe Flajolet, Analytic Combinatorics. Fridays, 10h30-12h30 and 14h30-16h30. The Book is available online here. Plan:
1. Symbolic methods, 11 September
2. Complex asymptotics, 18 September
3. Singularity analysis, 25 September
4. Limit distributions, 2 October (unusual location : salle 314)
Christian Krattenthaler, Tilings and plane partitions. Thursdays 10, 17 September and 15, 22 October, 10-12h and 14-16h. Unusual location on 15 October : salle 314. Lecture materials are available here. Plan:
The first part of this course will provide an introduction to the enumerative theory of (mainly domino and rhombus) tilings and plane partitions. The second part will introduce into recently developed theory addressing asymptotic properties of tilings.
Grégory Miermont, Random maps. Fridays 16, 23, 30 October and 13, 20 November 2009, 10h-12h and 14h-16h. Location on 23, 30 October and 13 November : salle 314. Plan:
Random maps, i.e random embedded graphs, can be considered as discrete models for random surfaces. In these lectures, we will show how continuum random surfaces can indeed be described as limits of uniform random quadrangulations and other families of maps. This is achieved by a probabilistic interpretation of bijective techniques initiated by Cori-Vauquelin and Schaeffer, allowing to encode maps with labeled trees. Known convergence results for random labeled trees, involving the celebrated Brownian tree and Brownian snake, can be used to obtain geometric properties of the associated random surfaces and their scaling limits. Part of the material of this course is presented in the following survey article.
1. Random trees and their scaling limits
2. The Schaeffer bijection and its generalizations
3. Scaling limits of random planar quadrangulations : construction and topology
4. Invariance principles for Boltzmann maps
5. Multi-pointed bijection and essential uniqueness of geodesics
Gordon Slade, Self-avoiding walks. Tuesdays 13, 20, 27 October and 10, 17, 24 November 2009, 10h30-12h and 14h-15h30. Plan:
The self-avoiding walk is a simple mathematical model that is difficult to analyse, and for which many of the most important problems remain unsolved. The model is important, in particular, in the theory of critical phenomena in statistical mechanics, where it provides a fundamental example. This course will discuss some old and some more recent results about the self-avoiding walk. Some specific topics treated in the course are:
- predictions for critical exponents, summary of unsolved problems and known results in all dimensions, some related models,
- proof of the Hammersley–Welsh bound on the number of self-avoiding walks,
- the lace expansion for self-avoiding walk: derivation, diagrammatic estimates, convergence in dimensions above 4 and consequences,
- functional integral representations for the self-avoiding walk, differential forms and “mixed bosonic-fermionic” integrals,
- introduction to the renormalisation group approach to the 4-dimensional self-avoiding walk.
An outline of some of the material to be covered in the course can be found in the following preliminary draft survey article.

Mini-courses

Mihai Ciucu, Dimer packings with gaps and electrostatics. Thursdays 19, 26 November and 3 December 2009, 10h30-12h30.
Robert Cori, A combinatorial point of view on maps. Thursdays 19, 26 November (14h-15h30) and Tuesday 15 December (11h-12h30). Topics:
Maps as pairs of permutations. Genus theorem. Product of commutators. Riemann-Hurwitz formula. Maps on non-orientable surfaces. Indecomposable permutations.
Bertrand Eynard, Explicit computation of numbers of maps, solution of Tutte equations for various types of maps of all topologies. Mondays 12, 19 and 26 October 2009, 14h30-16h30. Plan:
- Introduction to maps and Tutte's equations (and possibly introduction to formal matrix integrals)
- Loop equations, solution of the master loop equation, counting maps on a disc
- Solution of higher topologies loop equations: enumeration of maps of genus g
- Limits of large maps, double scaling limits, KdV, (p,q) Liouville gravity
- Generalizations: Ising model, O(n) model, Potts model
- Further generalizations: "counting Riemann surfaces", Weil-Peterson volumes of moduli spaces, intersection numbers, Gromov-Witten invariants
Emmanuel Guitter, Distance statistics in planar maps. Mondays 12, 19 and 26 October 2009, 10h30-12h30. Slides
Planar maps, which are graphs embedded in the sphere, arise in various domains of physics as discrete models for fluctuating surfaces, for instance in the context of two-dimensional quantum gravity, string theory or membrane physics. They raise many interesting combinatorial and probabilistic questions. These lectures will focus more precisely on the distance statistics in planar maps and address questions intended to reveal the metric structure of large maps: What is the profile of distances between two random points in a large map ? How many geodesic (shortest) paths link two given points in the map ? What does a geodesic triangle between three points look like ?

We will answer these questions exactly at the discrete level for some families of maps and translate these results into universal scaling laws for the so-called Brownian map, which is the continuous scaling limit of large maps.
1. Planar maps and well-labeled mobiles
2. The two-point function of planar quadrangulations and its scaling limit
3. Statistics of geodesics in planar quadrangulations
4. The three-point function of planar quadrangulations and its scaling limit
If time allows, we will also discuss: quadrangulations without multiple edges, integrability for even-valent maps.
Remco van der Hofstad, Mean-field behavior of percolation above the upper critical dimension. Wednesday 21 October (14h30-16h30), Wednesday 28 and Thursday 29 October 2009 (10h30-12h30 and 14h30-16h30). Slides
In the past decades, substantial progress has been made in the understanding of percolation in high dimensions. Indeed, the pioneering works of Aizenman, Barsky and Newman show that when the so-called triangle condition, a condition on the percolation connectivity function, holds, then critical percolation on Z^d when d is large behaves similarly as it does on a regular infinite tree. The seminal works of Hara and Slade subsequently proved that the triangle condition holds, thus turning the conditional results into theorems. In particular, we now know that there is no infinite cluster at criticality, and the cluster size tail behave as it does when the underlying graph is a tree. The reason for this, informally, is that when the dimension is high, the space is so vast, that far away pieces of the percolation cluster hardly interact. Therefore, geometry plays a less profound role, or “trivializes”, thus suggesting that for most questions the answer would be as for percolation on an infinite regular tree.

In these talks we will discuss different aspects of making this notion rigorous and review some recent developments. We shall discuss percolation on a tree and prove the relevant results using branching processes. We then proceed to study percolation in high dimensions, discuss the results and show how the triangle condition can be used to determine the asymptotics of the expected cluster size close to criticality. Then, we shall describe the main tool used in high dimensions, the lace expansion, and show how it can be used to prove results about the percolation connectivity function. The aim of these lectures is to demystify the lace expansion, by explaining in some detail how it can be used to prove properties of critical percolation.

After this introduction on high-dimensional percolation, we shall describe a few recent developments, namely, random walks on infinite critical percolation clusters, and percolation on high-dimensional tori and the role of boundary conditions. We shall close by recent results on a class of inhomogeneous percolation models on the complete graph, where the scaling behavior can be markedly different due to the inhomogeneity. We shall assume no prior knowledge on percolation, and all lectures can be understood separately.
- Lecture 1 - Critical percolation on the tree and in high dimensions: results.
- Lecture 2 - Critical percolation in high dimensions: the lace expansion.
- Lecture 3 - The Incipient Infinite Cluster and Random walks on it.
- Lecture 4 - Critical percolation on high-dimensional tori: Finite size scaling in percolation.
- Lecture 5 - Critical inhomogeneous percolation on the complete graph.
Des Johnston, Lattice paths and (P)ASEP. Wednesdays 18 and 25 November 2009, 10h30-12h30.
The matrix product ansatz provides an appealing method for solving the one dimensional asymmetric exclusion process (ASEP) and partially asymmetric exclusion process (PASEP), particularly with open boundaries. It provides a valuable exact solution for a non-equilibrium driven, diffusive system and illuminates many of the properties of such systems.

The ASEP and PASEP have proved of interest not just as physical models of driven, diffusive systems but also in a combinatorial context. In these lectures we outline how the matrix product ansatz naturally leads to a lattice path interpretation of the (P)ASEP normalization and via this to a succinct (re)derivation of the ASEP and PASEP phase diagram.

This is achieved by consideration of a “grand-canonical” normalization which can be easily summed explicitly in the case of the ASEP. The current author is too ignorant to have done this for the PASEP (and would appreciate the help of any audience in doing so) but a diversion into Flajolet's work on the use of continued fraction for lattice path generating functions still allows access to the phase diagram and other results. In addition, there are some intriguing links between the resolvents of the continued fractions which appear in this approach and finite dimensional representations of the PASEP algebra.

The original papers giving the matrix ansatz solution of the ASEP:

Exact solution of a 1D asymmetric exclusion model using a matrix formulation,
B. Derrida, M. R. Evans, V. Hakim and V. Pasquier, J. Phys. A: Math. Gen. 26 (1993) pp1493–1517

and PASEP:

Exact solution of a partially asymmetric exclusion model using a deformed oscillator algebra,
R. A. Blythe, M. R. Evans, F. Colaiori and F. H. L. Essler, J. Phys. A: Math. Gen. 33 (2000) pp2313–2332

provide very readable source material. I shall also shamelessly plagiarize (with the authors' permission) the very good review of the matrix ansatz method:

Nonequilibrium Steady States of Matrix Product Form: A Solver's Guide,
R. A. Blythe and M. R. Evans, J. Phys. A Math. Theor. 40 (2007) R333-R441.

A discussion of the relation of the PASEP to lattice paths and continued fractions can be found in:

Continued Fractions and the Partially Asymmetric Exclusion Process,
R. A. Blythe, W. Janke, D. A. Johnston and R. Kenna, J. Phys. A: Math. Theor. 42 325002 (2009).
Martin Loebl. Monday 28 and Tuesday 29 September 2009, 10h30-12h. Plan:
1. On the optimality of the Arf invariant formula for Ising partition function: recent joint work with Gregor Masbaum on the complexity of computing Ising partition function will be explained. Slides
2. q-binomial counting, graph and knot polynomials, permanents: some recent results connecting the graph and knot polynomials will be presented. Slides
Kirone Mallick, Bethe Ansatz for the Exclusion Process. Wednesdays 14 and 21 October 2009, 10h30-12h30.
The asymmetric simple exclusion process (ASEP) can be solved exactly using integrability techniques borrowed from the theory of quantum integrable systems. Although widely used by mathematical physicists, these methods are less known to combinatorists. We shall use the ASEP as a template to learn various aspects of the Bethe Ansatz; we shall show how spectral properties of the evolution matrix, current fluctuations and mappings to vertex models can be understood with the help of coordinate, fonctional and algebraic Bethe Ansatz.

Seminars

The trimester featured of course seminars given by participants. Their organization was kindly taken care of by Des Johnston.

Ross Kang, Degree-based partitions of random graphs. Tuesday 15 September, 11h.
Damien Simon, Bethe Ansatz for the open asymmetric exclusion process. Thursday 24 September, 10h.
Alain Lascoux, Link Patterns and K-L bases. Thursday 24 September, 14h.
Malwina Luczak, Glauber dynamics for the Ising model on the complete graph. Tuesday 29 September, 14h.
Jang Soo Kim, New interpretation for noncrossing partitions of classical types. Thursday 1 October, 10h.
Alan Sokal, Some wonderful conjectures (but almost no theorems) at the boundary between analysis, combinatorics and probability: The entire function $F(x,y) = \sum\limits_{n=0}^\infty \frac{x^n}{n!} y^{n(n-1)/2}$ , the polynomials $P_N(x,y) = \sum\limits_{n=0}^N \binom{N}{n} x^n y^{n(N-n)}$ , and the generating polynomial of connected graphs. Monday 9 November, 14h. Slides
Svante Janson, Susceptibility of random graphs. Monday 9 November, 16h.
Antti Kemppainen, Describing random curves by Loewner evolutions. Thursday 12 November, 10h.
Antti Kupiainen, Random Conformal Welding. Thursday 12 November, 14h.
Mark Dukes, Ascent sequences, (2+2)-free posets, pattern avoiding permutations, and related structures. Wednesday 18 November, 14h.
François David, KPZ relations and heat kernels. Tuesday 1 December, 11h.
Matthieu Josuat-Vergès, The PASEP partition function and Laguerre histories. Tuesday 1 December, 14h.

Last update of this page: 22 April 2015.