09:30 to 10:15 R Fernandez ([Rouen])Analyticity of the pressure of the hard-sphere gas The talk will present an improved estimate of the radius of analyticity of the pressure of the hard-sphere gas. It is obtained through a more careful estimation of the convergence of the corresponding cluster expansion, by exploiting Penrose identity for the coefficients of the expansion. INI 1 10:15 to 11:00 Integral equations for cluster expansion sums of polymer models with (soft) repulsion In this note, we derive integral equations for quantities a(P) defined as sums of cluster (Mayer) expansion terms taken over clusters "touching" a given polymer P. Using these quantities, many of the interesting properties of the model, like correlation decay can be evaluated. These relations have a slightly more complicated structure than usual exponential, "K.P." inequalities for quantities a(P) (developed successively by many authors: Cammarota, Seiler, Kotecky, Preiss, Dobrushin, Sokal, Scott, Fernandez, Ueltschi, and others) but they are equations. Their derivation employs quantities a(P,t) of auxiliary models with repulsions around P "softened" by an artificial parameter t, 0 < t < 1. Then at least some of the inequalities mentioned above can be interpreted as sufficient conditions (on the "smallness" of the complex polymer weights w(P)) for the convergence of the fixpoint method of solving such an integral equation. INI 1 11:00 to 11:30 Coffee 11:30 to 12:30 V Rivasseau ([Paris Sud])Introduction to non commutative field theory We recall some standard tools of ordinary field theory (Parametric representation, cluster expansions, renormalization) and the interesting related combinatoric aspects (tree matrix theorem, forest formulas, Hopf algebras...). We then outline the basic definitions of non commutative field theory and sketch how to generalize these tools to the non commutative case. This typically leads to even richer combinatoric problems and identities. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 Potts model, O(n) non-linear sigma-models and spanning forests In modern language, Kirchhoff Matrix-Tree Theorem (of 1847) puts in relation the (multivariate) generating function for spanning trees on a graph to the partition function of the scalar fermionic free field. A trivial corollary concerns rooted spanning forests and the massive perturbation of the free field. We generalize these facts in many respects. In particular, we show that a fermionic theory with a 4-fermion interaction gives the generating function for unrooted spanning forests, which are a limit of Potts Model for q -> 0. Remarkably, this theory coincides with the perturbative theory originated from a non-linear sigma-model with OSP(1|2) symmetry, which, in Parisi-Sourlas correspondence, is expected to coincide with the analytic continuation of O(n) model to n -> -1. The relation between spanning forests and the fermionic theory can be proven directly with combinatorial methods. However, the underlying OSP(1|2) symmetry leads to the definition of a subalgebra of Grassmann Algebra (the scalars under global rotations), with a set of surprising properties that quite simplify all the proofs. With some effort we can also generalize the whole derivation to a family of theories with OSP(1|2m) symmetry, with m a positive integer. INI 1 15:00 to 15:30 Tea 15:30 to 16:15 An extensor tree theorem and a Tutte identity for graphs with distinguished port edges Suppose Kirchhoff's laws for electricity are expressed by the rows of a matrix, each value of current flow and voltage (drop) along an edge is expressed by a separate variable, and those edges that model resistors each have a weight. Each non-resistor edge is distinguished as a port; each port's voltage and current relate the network to its environment. Each matrix row can be expressed as a linear combination of basis (co-)vectors. The basis vectors function as rank 1 extensors which can also be thought of as fermionic (anticommuting) Grassmann-Berezin variables. The (anticommutative) exterior product of the rows followed by the operation equivalent to eliminating the resistor current and voltage variables produces the extensor we will study. We prove that the resulting extensor obeys Tutte-like identities with deletion/contraction allowed only for the non-distinguished (resistor) edges, with anticommutative multiplication, and with sign-correction factors. We get as a result $\binom{2p}{p}$ different weigted matrix-tree theorems, where $p$ is the number of port edges. One application is a combinatorial but not bijective proof of Rayleigh's inequality alternative to that given by Choe. Our construction and the Tutte identities extend to arbitrary full-rank matrices, but the coefficients in the enumeration fail to be $\pm 1$ without unimodularity. We will also sketch analogies between our exterior algebraic formulation and the Grassmann-Berezin Calculus. INI 1 16:15 to 17:00 R Gurau ([Paris Sud])Parametric representation of non commutative quantum field theory We present the generalization of the parametric representation of ordinary quantum field theory to the non commutative case. Its derivation reposes largely on the use of Grassmann numbers, variables adapted to the computation of determinants. The non commutative generalization of Symanzik polynomials are found to be sums of positive terms indexed by new topological objects, called "bytrees" which will be discussed in detail. INI 1 18:45 to 19:30 Dinner at Wolfson Court
 09:30 to 10:15 J Imbrie ([Virginia])Forest-root formulas in statistical physics I will discuss forest-root formulas as interpolation identities in R^n and in C^n. These identities have applications in cluster expansions and Mayer expansions. They can be used to solve the combinatorics of branched polymers by relating them to negative-activity hard-core gases in lower dimensions. INI 1 10:15 to 11:00 J Magnen (École Polytechnique)Constructive field theory without tears We present a preliminary result of a programme aiming at simplifying the basic arguments of constructive (i.e. non-perturbative) field theory. It expresses the pressure of the $\phi^{4}$ model with cut-offs as a convergent sum of terms associated to trees whose vertices are non-perturbative loops. We give also the expression for a correlation function. INI 1 11:00 to 11:30 Coffee 11:30 to 12:30 Identities for fully packed loop configurations and semistandard tableaux I shall present joint work with Fabrizio Caselli, Bodo Lass and Philippe Nadeau, and work of Johan Thapper in direction of conjectures of Jean-Bernard Zuber on the enumeration of fully packed loop configurations with a fixed associated matching pattern. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:15 to 15:00 Counting partially directed walks in a symmetric wedge The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional recurrence for the generating function, and its solution. We consider a model of partially directed walks from the origin in the square lattice confined to a symmetric wedge defined by Y = ±X. We derive a functional equation for the generating function of the model, and obtain an explicit solution using a version of the Kernel method. This solution shows that there is a direct connection with matchings of an 2n-set counted with respect to the number of crossings, and a bijective proof has since been obtained. Related Links http://www.maths.qmul.ac.uk/~tp/papers/pub057.pdf - Partially directed paths in a wedge (van Rensburg; Prellberg; Rechnitzer)http://arxiv.org/abs/0712.2804v3 - Nestings of Matchings and Permutations and North Steps in PDSAWs (Rubey)http://arxiv.org/abs/0803.4233v1 - A Bijection Between Partially Directed Paths in the Symmetric Wedge and Matchings (Poznanovik) INI 1 15:00 to 15:30 Tea 15:30 to 16:15 Enumeration and asymptotics of random walks and maps In this talk, I want to give a brief survey of what I did in analytic combinatorics (=using generating functions to enumerate combinatorial structures, and then using complex analysis to get the asymptotics). This survey will be based on 3 kinds of equations which are often met in combinatorics, the way we solve them, and what kind of generic methods we use to get the full asymptotics/limit laws. By full asymptotics, I mean an expansion like $$f_n \sim C A^n n^\alpha + C' A^n n^(\alpha-1) + C'' A^n n^(\alpha-2) + \dots$$ where $A$ is the growing rate and $\alpha$ the "critical exponent" of the corresponding combinatorial structure. Namely, I will show that three combinatorial structures are "exactly solvable" : - a directed random walk model (using the kernel method and singularity analysis of algebraic functions), - random walks on the honeycomb Lattice (using an Ansatz and Frobenius method for D-finite functions), - question of connectivity in planar maps (using Lagrange inversion and coalescing saddle points, leading to a ubiquitous distribution involving the Airy function). This talk is based on an old work with Philippe Flajolet, Michèle Soria, and Gilles Schaeffer, and on work in progress with Bernhard Gittenberger. INI 1 16:15 to 17:00 A rosetta stone: combinatorics, physics, probability The purpose of this talk is to relate enumerative combinatorics, statistical physics, and probability. Specifically, it will explain how the theory of species of structures is exactly what is needed for the equilibrium statistical mechanics of particles. Thus, for example, the condition that the rooted tree equation has a finite fixed point is precisely equivalent to the Kotecky-Preiss condition used in the theory of cluster expansions. Furthermore, there is a fixed point equation for rooted connected graphs from which the rooted tree bound follows immediately. The talk will conclude with an indication about how the recent Fernandez-Procacci cluster expansion condition corresponds to an enriched rooted tree bound. Conclusion: Combinatorics and mathematical physics tell the same story, perhaps in different languages. INI 1 18:45 to 19:30 Dinner at Wolfson Court