Matching probabilities and loop models

In the last decade there has been a growing interest in the connection between combinatorics and statistical physics. This is caused by the remarkable discovery made by Alexander Razumov and Yuri Stroganov [1] in 2004, proven by Luigi Cantini and Andrea Sportiello [2] in 2011, of an unexpected connection between two loop models. Loop models are probability distributions for paths on a lattice. The paths are either closed or terminate on the boundary of the lattice. Here we consider two such models on very different geometries.

Fully Packed Loops on a finite grid. First off, consider a square grid of which the edges are traversed by paths such that each point of the grid is visited by precisely one path and such that every other external edge of the grid is covered by a path (Fig. 1). Such a decoration of the grid by paths is called a Fully Packed Loop (FPL) configuration.

Figure 1. An FPL configuration on a 3×3 square grid. The edges covered by a path are coloured red.

Two external edges are said to be matched by the FPL configuration if they are visited by the same path. In this way each FPL configuration gives rise to a pairwise matching of the occupied half of the external edges. In the example of Fig. 1 those external edges are labelled from 1 to 6, and the FPL configuration yields the following matching 1↔2, 3↔4, 5↔6.

What is the probability of a given matching of the occupied external edges, when each FPL configuration is equally likely? This is a counting problem: the probability is given by the number of FPL configurations that produce the chosen matching divided by the total number of FPL configurations. For example, on a 3×3 grid there are two FPL configurations producing the matching 1↔2, 3↔4, 5↔6 (Fig. 1 is one of them), while there are in total seven FPL configurations shown in Fig. 2. The probability to obtain the matching 1↔2, 3↔4, 5↔6 thus equals 2/7.

Figure 2. The seven FPL configurations on a 3×3
square grid.

Completely Packed Loop model on a half-infinite cylinder. Now consider non-intersecting paths on a square lattice on a half infinite cylinder, such that each edge is traversed once, and consequently each vertex is visited twice. It can be seen as being built up from the tiles shown in Fig. 3.

Figure 3. The two possible tiles of the CPL model.

This is called the Completely Packed Loop (CPL) model. As in the FPL model, all path configurations are equally probable. Fig. 4 shows a possible configuration of the top two layers. Again, each configuration determines a matching of the endpoints of the paths at the circular rim of the cylinder (see Fig. 5). Computing the matching probability in the CPL model is not a counting problem, since there are infinitely many CPL configurations (each equally probable).

Figure 4. The top two layers of a CPL configuration on a half-infinite cylinder.

The surprising connection. Razumov and Stroganov [1] discovered that the probability that a specific matching results from a randomly chosen CPL configuration on the cylinder is the same as the probability that it is induced by a randomly chosen FPL configuration on the square grid. The conjecture was proven by Luigi Cantini and Andrea Sportiello [2] but their proof did not provide a conceptual understanding of this surprising fact.

Figure 5. Inside view of a CPL configuration producing the matching 1↔2, 3↔4, 5↔↔6 at the circular rim.

Generalisations and future developments. In the meanwhile, the Razumov- Stroganov conjecture has been generalised in many ways, to different geometries, boundary conditions, and model rules. The guiding principle here is integrability, a very special feature of the models and their generalisations which, in case of the CPL model, is reflected by the remarkable property of the model that the matching probabilities are unaltered if the tile bias is changed, i.e. if one tile is favoured over the other. The generalisations have led to a host of interesting new conjectures of correspondences, and of explicit expressions for probabilities of certain events. Few of these have been proven.

The generalisations have led to a host of interesting new conjectures

A natural generalisation of the CPL model is obtained by allowing the tile bias to differ for each column on the half-infinite cylinder. The dependence of the corresponding matching probabilities on the column biases is governed by a wellknown set of equations in mathematical physics, called quantum Knizhnik-Zamolodchikov equations. It is hoped that cutting-edge techniques from mathematics and theoretical physics will demystify and generalise the mysterious connection between the matching probabilities. We are both actively involved in this exciting research area [3,4] and are currently co-supervising a PhD-project on this topic.

Acknowledgments: We thank Jim de Groot (Master’s student in Mathematics at the UvA) for his help in producing the figures. Ω

BERNARD NIENHUIS is Professor of Condensed Matter Theory at the Institute for Theoretical Physics, UvA.

JASPER STOKMAN is Professor of Lie Theory at the Korteweg-de Vries Institute, UvA.

References:

[1] A.V. Razumov, Yu.G. Stroganov, Theoret. and Math. Phys. 138, 333–337 (2004).

[2] L. Cantini, A. Sportiello, J. Combin. Theory Ser. A 118, 1549–1574 (2011).

[3] S. Mitra, B. Nienhuis, J. de Gier, M. T. Batchelor, JSTAT, P09010 (2004) [4] J.V. Stokman, B.H.M. Vlaar, J. Approx. Theory 197, 69 – 100 (2015).