IS09: Random partitions

date: 7/14/2025, time: 14:00-15:30, room: ICS 141

Organizer: Harriet Walsh (University College Dublin)

Chair: Harriet Walsh (University College Dublin)

Discrete \(N\)-particle systems at high temperature through Jack generating functions

Maciej Dołęga (Institute of Mathematics of the Polish Academy of Sciences)

We discuss random discrete \(N\)-particle systems, which can be also interpreted as random partitions, with the deformation (inverse temperature) parameter \(\theta\). We find necessary and sufficient conditions for the Law of Large Numbers as their size \(N\) tends to infinity simultaneously with the inverse temperature going to zero.

We apply the general framework to obtain the LLN for a large class of Markov chains of \(N\) nonintersecting particles with interaction of log-gas type, and the LLN for the multiplication of Jack polynomials, as the inverse temperature tends to zero. We express the answer in terms of novel one-parameter deformations of cumulants, and we discuss their relation with quantized free probability and continuous log-gas systems. Based on joint work with Cesar Cuenca.

Universality for random permutations

Slim Kammoun (université de Poitiers)

We consider a partition \(\lambda\), either deterministic or random, and sample a uniform permutation conditioned to have cycle structure \(\lambda\). We then study the shape of its image under the Robinson–Schensted correspondence. Under certain conditions on the total number of blocks of \(\lambda\) (or the number of blocks of size one), we establish a universal limiting behavior for the shape. This talk is based on the work \([2]\) and recent results by Dubach \([1]\).

Bibliography

\([1]\) Dubach, V. (2024). A geometric approach to conjugation-invariant random permutations. arXiv:2402.10116

\([2]\) Mohamed Slim Kammoun (2022). Universality for random permutations and some other groups. Stochastic Processes and their Applications, 147, 76–106. DOI

Random planar trees and the Jacobian conjecture

Elia Bisi (University of Florence)

A map \(F(x)=(F_1(x_1,\dots,x_n),\dots,F_n(x_1,\dots,x_n))\) from \(\mathbb{C}^n\) to itself is called a polynomial map if each \(F_i\) is a polynomial with complex coefficients in the variables \(x_1,\dots,x_n\). In 1939, Keller conjectured \([1]\) that every polynomial map \(F\) whose Jacobian determinant is a nonzero constant has a compositional inverse \(F^{-1}\) that is itself a polynomial map. This hypothesis, known as the Jacobian conjecture, is still one of the greatest unsolved problems of mathematics, and appears in Smale’s list \([2]\) of 18 open mathematical problems for the 21st century. The combinatorial approach to the Jacobian conjecture, developed more recently from the original context of algebraic geometry, is based on rephrasing the underlying concepts in terms of families of trees. We provide some new insights in this line of research, by using probabilistic methods, such as random trees, Markov chains, and branching processes. This approach also allows us to show that the high degree coefficients of \(F^{-1}\) are ‘small’, thereby proving an approximate version of the Jacobian conjecture.

Based on joint work with Piotr Dyszewski, Nina Gantert, Samuel G.G. Johnston, Joscha Prochno, and Dominik Schmid \([3]\).

Bibliography

\([1]\) O. H. Keller: “Ganze Cremona-Transformationen”, Monatsch. Math. Phys. 47, pp. 299–306 (1939).

\([2]\) S. Smale: “Mathematical problems for the next century”, The Mathematical Intelligencer 20:2 (1998).

\([3]\) E. Bisi, P. Dyszewski, N. Gantert, S. G. G. Johnston, J. Prochno, D. Schmid: “Random planar trees and the Jacobian conjecture”, arXiv:2301.08221 (2023+).