Event at Galileo Galilei Institute

Focus Week

Lattice models: exact methods and combinatorics

May 18, 2015 - May 22, 2015

Abstract

This event is one of the activities taking place at Galileo Galilei Institute, in Florence, in the framework of the two month workshop on Statistical Mechanics, Integrability and Combinatorics. It aims at bringing together mathematicians and physicists interested in problems at the interface between statistical mechanics, combinatorics and probability theory.

In this Focus Week, emphasis is put on lattice models, and their behaviour at a macroscopic level (scaling limits at criticality) and especially at a microscopic level (combinatorial and algebraic structures).

Organizers

Filippo Colomo (INFN, Florence)
Yacine Ikhlef (CNRS, UPMC-Paris 6)
Paul Zinn-Justin (CNRS, UPMC-Paris 6)

Contact
colomo@fi.infn.it

Main event
Statistical mechanics, integrability and combinatorics (Workshop) - May 11, 2015

Program Speakers Schedule

Talks

Date	Speaker	Title	Type	Useful Links
May 18, 2015 - 10:30	Guttmann Tony	Integrability, Solvability and Enumeration	Seminar	Abstract Integrability, Solvability and Enumeration There are a number of seminal two-dimensional lattice models that are integrable, but have only been partially solved, in the sense that only some properties are fully known (e.g. the two- dimensional Ising model, where the free-energy is known, but not the susceptibility). Alternatively, critical properties are known for some lattices but not others. For example, the critical point of the self-avoiding walk model is known rigorously for the honeycomb lattice, but not for other lattices. Similarly for the q-state Potts model and both bond and site percolation. The critical manifold of the former is known only for some lattices, likewise the percolation threshold is known only for some lattices. A range of numerical procedures exist, based on exact enumeration, or other numerical work, such as calculating the eigenvalues of transfer matrices, which, when combined with various structural invariants seem to give exact results in those cases that are known to be exact, but can be used to give increasingly precise estimates in those cases which are not exactly known. Reasons for this partial success are not well understood. In this talk I will describe four such procedures, and demonstrate their performance, and speculate on their partial success	Slides
May 18, 2015 - 11:30	Biane Philippe	Covering trees of the graph of covering trees	Seminar	Abstract Covering trees of the graph of covering trees The set of covering trees of a directed graph can be endowed with a natural graph structure covering the original graph. We show that the generating function of covering trees of this new graph exhibits a remarkable factorization property. This is joint work with Guillaume Chapuy.	Slides
May 18, 2015 - 15:00	Johansson Kurt	Random matrices and Aztec diamonds	Seminar	Abstract Random matrices and Aztec diamonds Random tilings, or dimer models on bipartite graphs, give rise to determinantal point processes. Natural scaling limits of these processes give rise to limiting processes and distributions of the same type as we see in random matrix theory and also in random growth models in the KPZ universality class, e.g the Airy process. I will give some background on this and in particular discuss it in connection with the two-periodic Aztec diamond, which is recent joint work with Sunil Chhita	Slides
May 18, 2015 - 16:30	Striker Jessica	Posets of alternating sign matrices and totally symmetric self-complementary plane partitions	Seminar	Abstract Posets of alternating sign matrices and totally symmetric self-complementary plane partitions Alternating sign matrices and totally symmetric self-complementary plane partitions are mysteriously equinumerous combinatorial objects with interesting connections to physics. In this talk, we discuss how encoding these objects as order ideals of partially ordered sets, or posets, yields a unifying framework for investigation. We study several actions on these objects from the poset perspective, including gyration on fully-packed loops, and show some surprising periodicity and orbit-average properties. We also give a permutation-case bijection and study the poset structures on both sides of this bijection.	Slides
May 19, 2015 - 09:00	de Gier Jan	A matrix product formula for Macdonald polynomials	Seminar	Abstract A matrix product formula for Macdonald polynomials I will discuss how an inhomogeneous version of the multi-species asymmetric simple exclusion process gives rise to a matrix product formula for Macdonald polynomials. This is work in collaboration with Luigi Cantini and Michael Whee	Slides
May 19, 2015 - 10:30	Corteel Sylvie	Combinatorics of exclusion processes with open boundaries	Seminar	Abstract Combinatorics of exclusion processes with open boundaries We study the combinatorics of the two-species asymmetric simple exclusion process with open boundaries studied among others by Masaru Uchiyama. We give a combinatorial interpretation of the partition function thanks to the Matrix Ansatz and the combinatorics of lattice paths. This brings us to a general positivity conjecture on Koornwinder moments. We also present a generalization of the staircase tableaux and TAT tableaux of Mandelshtam and Viennot which gives a combinatorial interpretation of the stationary distribution. This is joint work with Lauren Williams (Berkeley) and also with Olya Mandelshtam (Berkeley) for the tableaux parts	Slides
May 19, 2015 - 11:30	Cantini Luigi	Inhomogenous Multispecies TASEP on a ring	Seminar	Abstract Inhomogenous Multispecies TASEP on a ring In this talk I will present some results about a multispecies version of the TASEP, a model which describes the stochastic evolution of a system of particles of different species on a periodic oriented one dimensional lattice, where two neighboring particles exchange their positions with a rate which depends on their species. For some choice of these rates the Markov matrix turns out to be integrable and for the same choice the (unnormalized) stationary probability is conjectured to show remarkable positivity and combinatorial properties related to Schubert polynomials. I will discuss how integrability leads to an interesting algebraic structure underlying the problem, and allowing to prove some remarkable properties of the stationary measure and to give exact formulas for the stationary probability of some classes of configurations.	Slides
May 19, 2015 - 15:00	Beffara Vincent	Smirnov s parafermionic observable, at and away from criticality	Seminar	Abstract Smirnov s parafermionic observable, at and away from criticality Smirnov s parafermionic observable, together with the theory of discrete holomorphicity, was instrumental in proving the conformal invariance of the interfaces of the Ising model at its critical point; it exhibits a flavor of integrability at the critical temperature for all the random-cluster models, and in a few cases (namely the Ising model itself and the RC models for q>4 ) it can be studied directly at non-critical temperatures, where it provides precise information about the two- point function. I will present a few results in this direction, joint with H. Duminil-Copin and S. Smirnov	Slides
May 19, 2015 - 16:30	Weston Robert	Discrete Holomorphicity and Perturbed Conformal Field Theory	Seminar	Abstract Discrete Holomorphicity and Perturbed Conformal Field Theory I will introduce the concept of discrete holomorphicity for functions on a 2D lattice embedded into the complex plane. I will briefly discuss the important role that functions with this property have played in proving the existence and uniqueness of the continuum limit of certain models of statistical mechanics. I will then go on to discuss a systematic procedure for constructing discretely holomorphic operators in solvable statistical mechanical models in terms of the underlying quantum group symmetry of these models. This procedure will be illustrated for the Chiral Potts model. I will show how the resulting discrete holomorphicity conditions for non-critical models leads directly to an identification of these models with a corresponding perturbed conformal field theory.	Slides
May 20, 2015 - 09:30	Bleher Pavel	Dimer Model: Full Asymptotic Expansion of the Partition Function	Seminar	Abstract Dimer Model: Full Asymptotic Expansion of the Partition Function We obtain the full asymptotic expansion of the partition function of the 2D dimer model on the m x n lattice with free and cylindrical boundary conditions. We show that the asymptotic expansion goes over powers of 1/S , where S=(m+1)(n+1) for the free boundary conditions and S=(m+1)n for the cylindrical boundary conditions. The coefficients of the asymptotic expansion are expressed in terms of the Kronecker double series, the Dedekind eta function, and the Jacobi theta functions. Our calculations use the technique of the work of Ivashkevich, Izmailian, and Hu, in which the full asymptotic expansion is developed for the periodic boundary conditions. This is a joint work with Brad Elwood and Drazen Petrovic.	Slides
May 20, 2015 - 10:30	Panova Greta	Lozenge tilings and other lattice models from the viewpoint of symmetric functions	Seminar	Abstract Lozenge tilings and other lattice models from the viewpoint of symmetric functions What do lozenge tilings (a.k.a. plane partitions, dimer covers of the hexagonal lattice), alternating sign matrices (or the six-vertex model) and the dense loop model have in common? For one, their limiting behavior can be studied with the help of some "asymptotic" algebraic combinatorics. We develop methods to analyze normalized symmetric functions (Schur functions and more general Lie group characters), as the indexing partition converges to a limiting profile. We apply this analysis together with some combinatorial interpretations to study the limiting behavior of the integrable models listed above. In particular, we show that the positions of horizontal lozenges near a vertical flat boundary are distributed like the eigenvalues of GUE matrices, and this holds for a wide class of domains (including such with free boundary). These methods can also be used to establish the existence of limit shapes also for free boundary domains. We discover Gaussian distribution for some observables of the Alternating Sign Matrices, leading again to GUE eigenvalues for the positions of 1s near the border (result of V. Gorin). We also find the asymptotics for the [conjectural] expected value of the mean total current between two adjacent points in the dense loop model. Based on joint work with Vadim Gorin. We will also discuss limit behavior under non-uniform ( q ^volume) distributions, based on ongoing work with Jonathan Novak	Slides
May 20, 2015 - 11:40	Ciucu Mihai	Lozenge tilings with gaps in a 90 degree wedge domain with mixed boundary conditions	Seminar	Abstract Lozenge tilings with gaps in a 90 degree wedge domain with mixed boundary conditions We consider a triangular gap of side two in a 90 degree angle on the triangular lattice with mixed boundary conditions: a constrained, zig-zag boundary along one side, and a free lattice line boundary along the other. We study the interaction of the gap with the corner as the rest of the angle is completely filled with lozenges. We show that the resulting correlation is governed by the product of the distances between the gap and its three images in the sides of the angle. This provides evidence for a unified way of understanding the interaction of gaps with the boundary under mixed boundary conditions, which we present as a conjecture. Our conjecture is phrased in terms of the steady state heat flow problem in a uniform block of material in which there are a finite number of heat sources and sinks. This new physical analogy is equivalent in the bulk to the electrostatic analogy we developed in previous work, but arises as the correct one for the correlation with the boundary. The starting point for our analysis is an exact formula we prove for the number of lozenge tilings of certain trapezoidal regions with mixed boundary conditions, which is equivalent to a new, multi- parameter generalization of a classical plane partition enumeration problem (that of enumerating symmetric, self-complementary plane partitions).	Slides
May 21, 2015 - 09:00	Di Francesco Philippe	Physics and combinatorics of the octahedron equation: from cluster algebras to arctic curves	Seminar	Abstract Physics and combinatorics of the octahedron equation: from cluster algebras to arctic curves The octahedron equation is a non-linear system of recursion relations in discrete 2+1 dimensions, that first appeared in the context of generalized Heisenberg quantum spin chains (T-system). Some solutions of this equation also have a purely combinatorial interpretation as generalized Coxeter- Conway Frieze patterns. This same equation plays a central role in the seemingly unrelated purely combinatorial problem of enumeration of domino tilings of a plane square-shaped domain called the Aztec diamond. For large size, the tiling configurations display some drastic change of typical behavior between the corners of the domain, where it freezes in certain tiling patterns, and a central region in the bulk, with disorder. In the continuum limit of large size and small mesh, the separation of phases is along the so-called "arctic circle" inscribed inside the square domain. In this talk, we shall show that the octahedron equation is part of a combinatorial structure called cluster algebra, and how its exact solution may be rephrased in terms of non-intersecting lattice paths, and eventually domino tilings. Using the octahedron equation explicitly, we will proceed and determine various arctic curves depending on initial data. As predicted by Kenyon and Okounkov, we find in general new "facet" phases of the dimer model within some connected components of these curves. (Joint works with R. Kedem and R. Soto Garrido).	Slides
May 21, 2015 - 10:30	Kedem Rinat	Fusion products and q -Whittaker functions	Seminar	Abstract Fusion products and q -Whittaker functions The solution of the generalized Heisenberg model via Bethe ansatz comes with generalized fermionic combinatorics for the spectrum. Using this to compute linearized partition functions relates this model to conformal blocks of WZW theories and graded finite-dimensional representations of affine algebras, which are tensor products. In the simplest example, these are affine Demazure modules, and the limit as the size of the system becomes infinite are the well- known level-1 representations associated with the CFT. In this talk, I concentrate on the finite system, and explain why the linearized partition functions in the level-1 case are Whittaker functions for the quantum (finite) algebra, which satisfy the quantum difference Toda equation.	Slides
May 21, 2015 - 11:40	Okada Soichi	Generalized Cauchy determinant and Schur Pfaffian, and their applications	Seminar	Abstract Generalized Cauchy determinant and Schur Pfaffian, and their applications The Cauchy determinant and the Schur Pfaffian, together with their generalizations, play a fundamental role in combinatorics and representation theory. For example, such determinant/Pfaffian identities are used to evaluate Izergin-Korepin-type determinants/Pfaffians in enumeration problems of alternating sign matrices. In this talk, we present yet other generalizations of the Cauchy determinant and the Schur Pfaffian, and give their applications to symmetric function identities.	Slides
May 21, 2015 - 15:00	Smirnov Fedor	On the space of quasi-local operators for the Fateev-Zamolodchikov model	Seminar	Abstract On the space of quasi-local operators for the Fateev-Zamolodchikov model We consider the Fateev-Zamolodchikov model (integrable anisotropic spin chain of spin 1). We show that the "good" basis of quasi-local operators is created by two doublets of fermions and one triplet of bosons. The latter constitute the Kac-Moody algebra of level one. This is to be compared with the spin 1/2 case for which we had one doublet of fermions.	Slides
May 21, 2015 - 16:30	Gohmann Frank	Low-temperature spectrum of correlation lengths of the XXZ chain in the anti-ferromagnetic massive regime	Seminar		Slides
May 22, 2015 - 09:00	Rosengren Hjalmar	Towards combinatorics of elliptic lattice models	Seminar	Abstract Towards combinatorics of elliptic lattice models Much of the recent progress on combinatorics of solvable lattice models is concerned with the six- vertex model and XXZ spin chain. For the more general eight-vertex model and XYZ spin chain, which are naturally parametrized by elliptic functions, progress has been slower. Various quantities related to the models are described in terms of polynomials with positive integer coefficients, but a combinatorial explanation for this phenomenon is still lacking. On the other hand, new intriguing features appear in the elliptic case, such as relations to the Painlevé VI equation already at the finite lattice. In the present talk, I will try to give an overview of the current knowledge, discussing the work of myself and others (V. Bazhanov, V. Mangazeev, A. Razumov, Yu. Stroganov, P. Zinn- Justin).	Slides
May 22, 2015 - 10:30	Kytola Kalle	Boundary correlation functions with a hidden quantum group	Seminar	Abstract Boundary correlation functions with a hidden quantum group In the scaling limit, critical lattice model correlation functions satisfy partial differential equations of conformal field theories. In this talk, we present a method of constructing boundary correlation functions using a hidden quantum group. Solutions with desired asymptotic behavior are found by solving a problem in representations of the quantum group. Two applications to random curves will be considered: multiple SLE pure partition functions and boundary visit amplitudes of chordal SLEs. The talk is based on joint work with Eveliina Peltola ([arXiv:1408.1384] and in preparation), and with Niko Jokela and Matti Järvinen ([arXiv:1311.2297])	Slides
May 22, 2015 - 11:30	Delfino Gesualdo	Phase separation, interfaces and wetting in two dimensions	Seminar	Abstract Phase separation, interfaces and wetting in two dimensions Separation of phases with finite correlation length and the same free energy is a common phenomenon. For long time the lattice integrability of the Ising model in two dimensions has provided the only exact result for the scaling limit of the order parameter profile. It has been recently realized that field theory yields a series of exact results for the different universality classes in two dimensions, including order parameter profiles, interface structure and wetting properties, both in the bulk and at boundaries. This is joint work with J. Viti ( Phase separation and interface structure in two dimensions from field theory , J. Stat. Mech. (2012) P10009) and with A. Squarcini ( Exact theory of intermediate phases in two dimensions, Annals of Physics 342 (2014) 171, and Phase separation in a wedge. Exact results , PRL 113 (2014) 066101	Slides