Dia: Dimecres 5 de març de 2025
Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.
A càrrec de: Misha Hlushchanka, University of Amsterdam
Títol: The independence polynomial on recursive graph sequences - the dynamical perspective
Resum: The distribution of the zeros of partition functions on graphs are intimately related to the analyticity of physical quantities and their phase transitions. In their pioneering work in the 1950s, Lee and Yang proved that the free energy per site of the cubic lattice is analytic at a given positive real parameter, provided that the complex zeros of the partition functions for a sequence of finite graphs converging to the lattice avoid a neighborhood of this parameter.
In the last decade, a special focus was put on graph sequences that do not converge to a regular lattice but are instead defined recursively. Examples of such recursive graph sequences include hierarchical lattices, Cayley trees, Sierpinski gasket graphs, and various self-similar Schreier graphs. Significant progress has been made in studying the partition functions of the Ising, Potts, and Hard-Core models on these graphs (with the latter two corresponding to the chromatic and independence graph polynomials, respectively). One main advantage of working with recursive sequences of graphs is that the underlying recursion naturally induces an iterative system on the level of partition functions, often given in terms of rational maps in one or several (complex) variables. By analyzing the dynamical behavior of these iterative systems, we can gain insights into the properties of the respective graph polynomials, such as phase transitions and computational complexity.
In our current work with Han Peters (University of Amsterdam), we attempt to establish a unified framework for recursive graph sequences in the Hard-Core model setting. We start with an arbitrary graph with k marked vertices. At each recursive step, we construct a new graph by taking n copies of the previous graph and connecting these copies along the marked vertices (according to a specified rule). The dynamical systems that emerge from the respective independence polynomials are represented by homogeneous polynomials of degree n in 2k variables. Somewhat surprisingly, it turns out that these systems can be successfully analyzed in this general context. In the talk, I will report on our results on the structure of the zero sets of the independence polynomials for these recursive graph sequences, particularly highlighting the absence of phase transitions.
Last updated: Tue Mar 4 11:28:09 2025