 # Statistical mechanics and combinatorics: February 15 – April 15, 2016

Statistical mechanics and combinatorics

Organized by Pavel Bleher, Vladimir Korepin, and Bernard Nienhuis
February 15 – April 15, 2016

The purpose of the program is to relate physics and mathematics, and more specifically, statistical mechanics, algebraic combinatorics, and random matrices. The program will focus on the six-vertex model of statistical mechanics and related models, such as the XXZ spin chain, and loop models. The six-vertex model was introduced by Pauling in 1935mas a two dimensional version of a model for the hydrogen bonds in ice.

The bulk free energy and entropy in the six-vertex model were explicitly calculated for periodic boundary conditions by Lieb. Then other boundary conditions were studied. A very interesting case is the domain wallboundary conditions (DWBC). The six-vertex model provides an important ‘counterexample’ in statistical mechanics: the bulk free energy in the thermodynamic limit depends on boundary conditions. In particular, it is different for periodic and domain wall boundary conditions.

The partition function of the six-vertex model with DWBC in a finite box has been expressed, via the Yang-Baxter equations, in terms of a Hankel determinant. It can be furthermore expressed as a partition function of an ensemble of random matrices with a non-polynomial interaction. This expression was used via the powerful Riemann-Hilbert approach, the asymptotic behavior of the partition function of the six-vertex model with DWBC in different phase regions.

A remarkable observation was made by Razumov and Stroganov that concerns the six-vertex model at a special value of the parameter Delta=1/2, on afinite square grid, transformed into a loop model, i.e. a model of paths on the lattice that can end only on the boundary. In this case the paths visit all vertices exactly once, and all possible configurations are equally probable. This simple measure induces a probability measure on how the edges on the boundary are pairwise connected to each other. This is compared with another loop model placed on a half-infinite cylinder. In this case it is a loop model where the paths pass every edge once, every vertex twice and do not intersect. An equivalent six-vertex model has Delta=-1/2. Again every configuration is equally probable, and again this induces a probability measure on a pairing of the boundary edges. The observation of Razumov and Stroganov is that both probability measures of pairings of boundary edges are the same.
This observation was later proven by Cantini and Sportiello. The original observation has many generalizations on different geometries, but these have not been proven.

The six-vertex model with DWBC relates statistical mechanics to various problems of combinatorics: the statistics of alternating sign matrices, domino tilings, limiting shapes, nonintersecting lattice paths, loop models, and others. Limiting shape formulae in the six-vertex model were proposed recently. The six-vertex model has various generalizations to the eight-vertex model, higher-spin systems, coloring of a lattice, and others, which are important for applications. We plan on inviting leading experts, both physicists and mathematicians, workingon the combinatorics of six-vertex model and its generalizations.

All program talks are held in SCGP Seminar Room 313.

#### Talk Schedule

 Date & Time Speaker Title 2/15 at 2pm Fabio Franchini Spontaneous ergodicity breaking in invariant matrix models 2/19 at 1pm Bernard Nienhuis Polynomial qKZ equations for loop models

 Date & Time Speaker Title 2/22 at 2pm Ramis Movassagh A New Class of Exactly Solvable Quantum Spin Chain models 2/22 at 1pm Nikolay Bogoliubov Random walks, plan partitions and correlation functions of Heisenberg chain in the limiting cases

 Date & Time Speaker Title 2/29 at 2pm Alexei Tsvelik SU(2n) -invariant spin ladder as a perturbed integrable theory 3/4 at 1pm Gernot Akemann Products of random matrices – exact solution and universality

 Date & Time Speaker Title 3/7 at 2pm Senya Shlosman Roughening and ballistic transitions in the interacting dimers 3/9 at 10:30am Soichi Okada Pfaffian identities and Schur’s Q-functions 3/11 at 1pm Barry McCoy The Once and Future Ising Model

 Date & Time Speaker Title 3/21 at 2pm Robert Shrock Some Results on Chromatic and Potts/Tutte Polynomials including Zeros and Asymptotic Limits for Families of Graphs 3/25 at 1pm Pavel Bleher The mother body phase transition in the normal random matrix model

 Date & Time Speaker Title 3/28 at 2pm Ivan Kostov Izergin-Korepin determinant, clustering and Sutherland limit 4/1 at 1pm Olof Salberger Shor—Movassagh Model at Half Integer Spins

 Date & Time Speaker Title 4/4 at 2pm Vassily Gorbunov, U Aberdeen, Scotland Hidden symmetries in equivariant cohomology related to quantum groups 4/8 at 1pm Jean-Bernard Zuber, U Pierre Marie Curie, Paris 6, France Counting curves and knots and links

 Date & Time Speaker Title 4/11 at 2pm Jacopo Viti Inhomogeneous quenches and arctic curves in fermionic systems. 4/15 at 1pm Vladimir Korepin Statistical Mechanics and Combinatorics 4/15 at 2pm Tzu-Chieh Wei Density of Yang-Lee zeros in the thermodynamic limit from tensor network methods