Conference Program

A partition identity is a theorem stating that for all $n$, the number of partitions of $n$ satisfying some conditions equals the number of partitions of $n$ satisfying some other conditions. For example, the Rogers-Ramanujan identities state that for all $n$, there are as many partitions of $n$ where parts differ by at leat two as partitions of $n$ into parts congruent to $1$ or $4$ modulo $5$. In terms of generating functions, partition identities can often be expressed as q-series identities in the form sum=product. The Andrews-Gordon identities are among the most important $q$-series and partition identities, and generalise the Rogers-Ramanujan identities. Interestingly, while the product side of their $q$-series version is clearly the generating function for partitions with congruence conditions, it is far from obvious that the sum side is the generating function for the partitions with frequency conditions that appear in the combinatorial version. It was originally proved by George Andrews using recurrences, and then bijectively by Ole Warnaar using particle motion, which can be seen as a bijection on lattice paths. In this talk, we will explain and generalise the particle motion approach. We will show that it can also be applied to the sum side of Bressoud's identity and that, using the Andrews-Gordon and Bressoud identities as starting points, many known and new identities can be proved. This is based on joint work with Jihyeug Jang, Frédéric Jouhet and Isaac Konan.

Motivated by previous work of Lazar and Linusson, I consider the refined enumeration of set-valued tableaux of two-rowed shapes. I shall derive formulae for the number of these tableaux, keeping track of the total number of entries, the number of entries in the first row, and the number of entries in the second row. Key in the proofs is a bijection with two-coloured Motzkin paths (which is already present in the work of Lazar and Linusson) followed by generating function computations and coefficient extraction helped by the Lagrange inversion formula.

We consider supercritical branching random walks (BRW) on infinite graphs, and we investigate the limit behavior of the population distributions. In particular, we look at the speed and the asymptotic entropy of such processes. Both these quantities describe non-trivial behavior at infinity of the BRW. We show that the entropy exhibits a phase transition depending on the underlying random walk. This talk is based on past and ongoing works in collaboration J. Brieussel, R. Kaiser, M. Klötzer, K. Kolesko, and H. Oppelmayer.

Enumerative and probabilistic properties about the set of walks on a graph can reveal important algebraic and combinatorial information about the graph and any object it encodes. In this talk we consider the types of equations satisfied by series related to excursions on graphs, taking both analytic and combinatorial perspectives. Looking at examples from Cayley graphs and families of self-similar graphs we try to identify hallmarks of algebraicity and differential transcendence. The context provides an excellent illustration of dichotomy results for order 1 iterative equations: solutions to these equations are either algebraic or differentially transcendental. (Work done in collaboration with Lucia Di Vizio, Gwladys Fernandes, and Yakob Kahane).

Sample abstract text for Igor Pak. Say, you are counting certain combinatorial objects. What is the range of the numbers you get? Do you ever get coincidences? If yes, how many? How hard is it to tell if you have a coincidence? I will discuss the coincidence problem for counting domino tilings, spanning trees and independent sets in graphs, linear extensions in posets, permutation patterns, and Young tableaux. I will also make a number of curious counting conjectures.

This talk will review recent results on the algebraic description of special functions and on the connections between the associated bispectral algebras and different areas of mathematics and mathematical physics.

Examples and perspectives will involve orthogonal polynomials, biorthogonal rational functions, algebraic combinatorics and exactly solvable quantum models.

The study of critical phenomena in lattice statistical mechanical models such as percolation has a long history in both physics and mathematics. A central problem is to derive the asymptotic behaviour of a model's two-point function, which reveals the values of critical exponents that govern the universal behaviour of the model at and near its critical point. Proofs are typically available only for dimension two, or above an upper critical dimension where mean-field behaviour is observed. The talk presents a general theorem providing the asymptotic decay of the solution to convolution equations of a certain sort. It applies to the lattice Green function (random walk in any dimension d), the generating function for the number of n-step self-avoiding walks from 0 to x (d at least 5), the probability that 0 and x are connected in a percolation cluster (high enough d). The general theorem gives a precise asymptotic formula for the decay of the two-point function in these three applications, which remains valid both in the subcritical regime (Ornstein-Zernike decay) and also at and near the critical point. It exhibits, in a unified and general way, the crossover from subcritical to critical decay. This is joint work with Yucheng Liu (Peking University).

The conference traditionally honors members of the lattice-path community who have sadly passed away since the previous edition. On this occasion, we will say a few words about the work of Charalambos A. Charalambides, Adrienne Kemp, Guy Louchard, Jean-Guy Penaud, and Renzo Sprugnoli.

A great many combinatorial structures are counted by generating functions satisfying a composition scheme $F(z)=G(H(z))$, also known as a Gibbs partition model. I will discuss critical schemes, where the corresponding asymptotic analysis becomes challenging because the generating functions G and H are simultaneously singular. These schemes are naturally related to lattice paths, random walks and a lot of other combinatorial structures. For an extension of the form $F(z,u)=G(u H(z))M(z)$, a rich variety of limit laws is obtained, involving mixed Poisson-, Boltzmann-, and generalized Mittag-Leffler distributions.

An enriched model of this scheme links to to various phenomena observed in the combinatorial and statistical physics literature in the context of q-enumeration. In particular, an application is given to zero contacts in non-intersecting lattice paths called watermelons. Furthermore, I will present a new regime for critical schemes related to families of lattice paths in cube, including simple walks and three-dimensional Delannoy walks. Finally, connections between lattice paths enumeration, card guessing procedures and lattice paths in the quarter plane are presented.