# Workshop Programme

## for period 21 - 25 April 2008

### Statistical-Mechanics and Quantum-Field-Theory Methods in Combinatorial Enumeration

21 - 25 April 2008

Timetable

 Monday 21 April 08:30-09:55 Registration 09:55-10:00 Welcome - David Wallace 10:00-11:00 Eynard, B (IPhT) Partitions, matrix models, and geometry Sem 1 Sums of partitions with the Plancherel measure appear in various problems of statistical physics and mathematics (crystal growth, point processes, Gromov-Witten invariants). We rewrite such sums as matrix integrals, which allow to compute their large size expansion to all orders. The coefficients in the expansion are geometric objects called symplectic invariants of spectral curves. This makes a link between combinatorics of partitions, matrix models, algebraic geometry, integrability, quantum field theory and topological string theory. 11:00-11:30 Coffee 11:30-12:30 Le Gall, J-F (Paris-Sud) The continuous limit of random planar maps Sem 1 Planar maps are graphs embedded in the two-dimensional sphere $S^2$, considered up to continuous deformation. They have been studied extensively in combinatorics, but they also have significant geometrical applications. Random planar maps have been used as models of random geometry in theoretical physics. Our goal is to discuss the convergence of rescaled random planar maps viewed as random metric spaces. More precisely, we consider a random planar map $M(n)$, which is uniformly distributed over the set of all planar maps with $n$ vertices in a certain class. We equip the set of vertices of $M(n)$ with the graph distance rescaled by the factor $n^{-1/4}$. We then discuss the convergence in distribution of the resulting random metric spaces as $n\to\infty$, in the sense of the Gromov-Hausdorff distance between compact metric spaces. This problem was stated by Schramm in his 2006 ICM paper, in the special case of triangulations. In the case of bipartite planar maps, we first establish a compactness result showing that a limit exists along a suitable subsequence. We then prove that this limit can be written as a quotient space of the so-called Continuum Random Tree (CRT) for an equivalence relation which has a simple definition in terms of Brownian labels assigned to the vertices of the CRT. This limiting random metric space had been introduced by Marckert and Mokkadem and called the Brownian map. It can be viewed as a Brownian surface'' in the same sense as Brownian motion is the limit of rescaled discrete paths. We show that the Brownian map is almost surely homeomorphic to the sphere $S^2$, although it has Hausdorff dimension $4$. Furthermore, we are able to give a complete description of the geodesics from a distinguished point (the root) of the Brownian map, and in particular of those points which are connected by more than one geodesic to the root. As a key tool, we use bijections between planar maps and various classes of labeled trees. Related Links http://www.dma.ens.fr/users/legall/ - contains links to papers related to the talk 12:30-13:30 Lunch at Churchill College 14:00-15:00 Bernardi, O (CNRS) A bijection for covered maps on orientable surfaces Sem 1 A map of genus g is a graph together with an embedding in the orientable surface of genus g. For instance, plane trees can be considered as maps of genus 0 and unicellular maps (maps with a single face) are a natural generalisation of plane trees to higher genus surfaces. In this talk, we consider covered maps, which are maps together with a distinguished unicellular spanning submap. We will present a bijection between covered maps of genus g with n edges and pairs made of a plane tree with n edges and a bipartite unicellular map of genus g with n+1 edges. This bijection allows to recover bijectively some very elegant formulas by Mullin and by Lehman and Walsh. We will also show that our bijection generalises a bijection of Bouttier, Di Francesco and Guitter (which, in turns, generalises a previous bijection of Schaeffer) between bipartite maps and some classes of labelled trees. 15:00-15:30 Tea 15:30-16:30 Bouttier, J (CEA Saclay) Vacancy localisation in the square dimer model, statistics of geodesic in large quadrangulations Sem 1 I shall present two independent projects carried out last year, illustrating the interests of a typical combinatorial statistical mechanist. - I first consider the classical dimer model on a square lattice with a single vacancy, and address the question of the possible motion of the vacancy induced by dimer slidings. Adapting a variety of techniques (Temperley bijection, matrix tree theorem, finite-size analysis, numerical simulations), we find that the vacancy remains localized albeit in a very weak fashion, leading to non-trivial diffusion exponents. [joint work with M. Bowick, E. Guitter and M. Jeng] - I next consider the statistical properties of geodesics in large random planar quadrangulations. Introducing "spine trees" extending Schaeffer's well-labeled tree construction, we obtain in particular the generating function for quadrangulations with a marked geodesic. We deduce exact statistics for large quadrangulations both in the "local" and "scaling" limits. [joint work with E. Guitter] 16:30-17:30 Majumdar, S (CNRS) Large deviation of the top eigenvalue of a random matrix Sem 1 The statistical properties of the largest eigenvalue of a random matrix are of interest in diverse fields ranging from disordered systems to string theory. In this talk I'll discuss some recent developements on the theory of extremely rare fluctuations of the largest eigenvalue and its various applications. Related Links http://fr.arxiv.org/abs/cond-mat/0609651 - paper published in Phys. Rev. Lett., 97, 160201 (2006)http://fr.arxiv.org/abs/0801.1730 - to appear in Phys. Rev. E (2008)http://fr.arxiv.org/abs/cond-mat/0701371 - published in J. Phys. A: Math. Theor. 40(16) (2007) 4317-4337 17:30-18:30 Welcome Wine Reception 18:30-19:00 Dinner at Churchill College (Residents only)
 Tuesday 22 April 09:00-10:00 Razumov, A; Stroganov, Y (Institute for High Energy Physics) Three colour statistical model with 'domain wall' boundary conditions Sem 1 In 1970 Baxter considered the statistical three-coloring lattice model satisfying toroidal boundary conditions. He used an appropriate version of the Bethe anzats and found the partition function of the model in the thermodynamic limit. We consider the same model but with specific boundary conditions. We find a functional equation satisfied by the partition function. A similar equation for the six vertex statistical model can be solved, and its solution can be used to prove the refined alternating-sing matrix conjecture. 10:00-11:00 Pasquier, V (IPhT) Alternating sign matrices from a physicist point of view Sem 1 There has recently been a convergence of interests between combinatorics and physics through the observation due to Razumov and Stroganov that the components of the ground state wave function of the 6-vertex (at a special value of the anisotropy parameter) model enumerate alternating sign matrices. This correspondence is still mysterious. I shall present my approach to the problem which uses the (Macdonald) polynomial representation of affine-Hecke algebras and points towards a connection with the Quantum Hall Effect. 11:00-11:30 Coffee 11:30-12:30 de Gier, J (Melbourne) Algebraic structure of the q-Knizhnik-Zamolodchikov equation on a segment, partial sums and punctured plane partitions Sem 1 We show that solutions of the qKZ equation for the link pattern representation of the Temperley-Lieb algebra have the same structure as the canonical basis defined by Lusztig in tensor products of representation modules of U_q(sl_2). This structure gives in a natural way rise to consider partial sums over qKZ solutions. In the context of the Razumov-Stroganov conjecture we show that such partial sums over qKZ solutions of level one are related to weighted transpose complement cyclically symmetric plane partitions with a hole. Related Links http://www.ms.unimelb.edu.au/~degier - Homepage 12:30-13:30 Lunch at Churchill College 14:00-15:00 Lascoux, A (CNRS) Gaudin functions of any order Sem 1 The Gaudin-Izergin-Korepin determinant plays a fundamental role in the 6-vertex model. We consider a more general determinant det( 1/(x-y)(x-ty)..(x-t^r y)), where x,y are indeterminates in two sets of the same cardinality and r is a fixed integer (the Izergin-Korepin determinant is the case r=1). We give specialization properties, as well as a link with Macdonald polynomials. Related Links http://arXiv:0709.1635 - detailed version 15:00-15:30 Tea 15:30-16:30 Okada, S (Nagoya) Multivariate generalisation of Hankel determinants of Catalan numbers and middle binomial coefficients Sem 1 The Hankel determinants of Catalan numbers and middle binomial coefficients are evaluated as products of simple factors. In this talk, I present multivariate determinant identities involving classical group characters, which reduce to the Hankel determinants of Catalan numbers and middle binomial coefficients if all the variables are equal to 1. 18:30-19:00 Dinner at Churchill College (Residents only)
 Wednesday 23 April 09:00-10:00 Smirnov, S (Geneva) Conformal invariance and universality in the 2D lsing model Sem 1 We will show that on a lage class of planar graphs, Ising model has a conformally invariant scaling limit at critical temperature. Joint work with Dmitry Chelkak (St.Petersburg State University) 10:00-11:00 Nienhuis, B (Amsterdam) Entanglement in the XXZ chain Sem 1 We take an antiferromagnetic XXZ chain in its ground state, and consider a small sequence of sites. The spins on these sites are entangled with the other spins. We show results for the entanglement entropy as a function of the length of the segment and the length of the chain. 11:00-11:30 Coffee 11:30-12:30 Jacobsen, JL (Ecole Normale Sup\'erieure) Exact valence bond entanglement entropy in the XXZ and related spin chains Sem 1 Valence bond entanglement entropy has recently been proposed as an alternative to the well-known von Neumann entropy, measuring the information density in the ground state of spin chains. It is the mean value of the number N of valence bonds (spin singlets) connecting a subsystem of length L >> 1 to the remainder of the chain. We determine exactly the complete probability distribution of N in the XXZ spin chain, as well as in other related spin chains. At the combinatorial point (bond percolation) we conjecture exact expressions of that same distribution for finite values of L and the chain length. 12:30-13:30 Lunch at Churchill College 14:00-15:00 Viennot, X (LaBRI/CNRS/Bordeaux 1) Alternative tableaux, permutations and partially asymmetric exclusion process Sem 1 We introduce a new combinatorial object called "alternative tableau". This notion is at the heart of different topics: the combinatorics of permutations and of orthogonal polynomials, and in physics the model PASEP (partially asymmetric exclusion process). The model PASEP have been recently intensively studied by combinatorists, in particular giving combinatorial interpretations of the stationary distribution (works of Brak, Corteel, Essam, Parvianinen, Rechnitzer, Williams, Duchi, Schaeffer, Viennot, ...). Some interpretations are in term of the so-called "permutation tableaux", introduced by Postnikov, followed by Steingrimsson and Williams, in relation with some considerations in algebraic geometry (total positivity on the Grassmannian). Permutations tableaux have been studied by Postnikov, Steingrimsson, Williams, Burstein, Corteel, Nadeau and various bijections with permutations have been given. The advantage of introducing alternative tableaux is to give a complete symmetric role for rows and columns. We give a bijection between the two kinds of tableaux and a direct bijection between permutations and alternative tableaux. We also give combinatorial interpretation of the stationary distribution of the PASEP in term of alternative tableaux. Then we show the relation between alternative tableaux and the combinatorial theory of orthogonal polynomials developed by Flajolet and the author. In particular the Françon-Viennot bijection between permutations and "Laguerre histories" (i.e. some weighted Motzkin paths) plays a central role here and enable us to construct another bijection between permutations and alternative tableaux. This last bijection is in the same vein as the construction of the classical Robinson-Schensted-Knuth correspondence by "local rules" as originally defined by Fomin. The bijection can also be presented in analogy with Schützenberger "jeu de taquin". We finish the talk by giving "la cerise sur le gâteau": the surprising connection between the two bijections relating permutations and alternative tableaux. Related Links http://www.labri.fr/perso/viennot/ - the home page of the author 15:00-15:30 Tea 15:30-16:30 Mallick, K (IPhT, Saclay) The asymmetric exclusion process: an integrable model for non-equilibrium statistical mechanics Sem 1 The Asymmetric Simple Exclusion Process (ASEP) plays the role of a paradigm in Non-Equilibrium Statistical Mechanics: it is one of the simplest interacting N-body systems far from equilibrium that can be solved analytically. By using the Bethe Ansatz, we calculate the spectral gap of the model and predict global spectral properties such as the existence of multiplets. We then discuss the fluctuations of the current in the stationary state. Finally, we explain that the stationary state of the ASEP and of some of its generalizations with multiple classes of particles has an underlying combinatorial structure that leads naturally to a matrix product representation. 18:30-19:00 Dinner at Churchill College (Residents only)
 Thursday 24 April 09:00-10:00 Reshetikhin, N (UC Berkeley, California) Dimer partition functions on surface graphs of higher genus Sem 1 10:00-11:00 Wilson, DB (Microsoft Research) Boundary partitions in trees and dimers Sem 1 Given a finite planar graph, a grove is a spanning forest in which every component tree contains one or more of a specified set of vertices (called nodes) on the outer face. For the uniform measure on groves, we compute the probabilities of the different possible node connections in a grove. These probabilities only depend on boundary measurements of the graph and not on the actual graph structure, i.e., the probabilities can be expressed as functions of the pairwise electrical resistances between the nodes, or equivalently, as functions of the Dirichlet-to-Neumann operator (or response matrix) on the nodes. These formulae can be likened to generalizations (for spanning forests) of Cardy's percolation crossing probabilities, and generalize Kirchhoff's formula for the electrical resistance. Remarkably, when appropriately normalized, the connection probabilities are in fact integer-coefficient polynomials in the matrix entries, where the coefficients have a natural combinatorial interpretation. A similar phenomenon holds in the so-called double-dimer model: connection probabilities of boundary nodes are polynomial functions of certain boundary measurements, and as formal polynomials, they are specializations of the grove polynomials. Upon taking scaling limits, we show that the double-dimer connection probabilities coincide with those of the contour lines in the Gaussian free field with certain natural boundary conditions. These results have direct application to connection probabilities for multiple-strand SLE_2, SLE_8, and SLE_4. Related Links http://arxiv.org/abs/math/0608422 - arXiv version 11:00-11:30 Coffee 11:30-12:30 Ciucu, M (Indiana) Dimer packings with gaps and electrostatics: boundary interactions Sem 1 In earlier work we determined the asymptotics of the correlation function of a collection of gaps in a sea of dimers. This turned out to be governed by the laws of two dimensional electrostatics. In this talk we consider dimer systems that cover a half-plane, and determine the interaction of gaps in this system with the boundary. We analyze the cases of constrained and free boundary conditions. They both lead to analogs of the method of images from electrostatics. 12:30-13:30 Lunch at Churchill College 14:00-15:00 Krattenthaler, C (Wien) Exact enumeration of plane partitions and rhombus tilings Sem 1 Problems of exact enumeration of plane partitions and rhombus tilings have appeared frequently in connection with the Razumov-Stroganov conjecture and related conjectures. In this talk, I shall discuss a miscellany of such problems, older and newer, relations in between, and the main ideas to solve such problems. 15:00-15:30 Tea 15:30-16:30 Boutillier, C (Université Pierre et Marie Curie) The bead model Sem 1 19:30-23:00 Conference Dinner at Jesus College (in the Upper Hall)
 Friday 25 April 09:00-10:00 Wu, FY (Northeastern) Theory of electric networks: the two-point resistance and impedance Sem 1 We present a new formulation of the determination of the resistance and impedance between two arbitrary nodes in an electric network. The formulation applies to networks of finite and infinite sizes. An electric network is described by its Laplacian matrix L, whose matrix elements are real for resistors and complex for impedances. For a resistor network the two-point resistance is obtained in terms of the eigenvalues and eigenvectors of L, and for an impedance network the two-point impedance is given in terms of those of L*L. The formulation is applied to regular lattices and non-orientable surfaces. For networks consisting of inductances and capacitances, the formulation predicts the occurrence of multiple resonances. Related Links http://www.physics.neu.edu/Department/Vtwo/faculty/wu%20files/FYWu_Publications041001.htm - Papers #210 and #215 in the list 10:00-11:00 Guttmann, A (Melbourne) Prudent and quasi-prudent self-avoiding walks and polygons Sem 1 Prudent self-avoiding walks and quasi-prudent self-avoiding walks are proper subsets of self-avoiding walks in dimension 2 or greater. Prudent SAW are not allowed to take a step in a direction which, if continued, would encounter a previously visited vertex. Quasi-prudent walks are self-avoiding walks where a step to a neighbouring vertex v can only be taken if there is a prudent way to escape from v (in other words, if v can be seen from infinity). Polygon versions of the walks can be defined as walks (prudent or quasi-prudent) which end at a vertex adjacent to their starting point. A variety of results, both rigorous and numerical, will be given for these models, mainly for two-dimensional walks, but we also have some preliminary results for walks on a three-dimensional lattice. 11:00-11:30 Coffee 11:30-12:30 Prellberg, T (QMUL) Counting lattice paths with the kernel method Sem 1 Models of directed paths have been used extensively in the scientific literature to model linear polymers. In this talk we examine directed path models of a linear polymer in various confining geometries. We solve these models by showing that the generating function satisfies a functional equation and deriving formal solutions by using the kernel method. While some generating functions are rational or algebraic, it turns out that in some interesting cases the generating functions are not differentiably finite. 12:30-13:30 Lunch at Churchill College 18:30-19:00 Dinner at Churchill College (Residents only)