Generalized duality for graphs on surfaces and the signed Bollobás–Riordan polynomial

AbstractWe generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed Bollobás–Riordan polynomials of dual graphs. This relation unifies various recent results expressing the Jones polynomial of links as specializations of the Bollobás–Riordan polynomials

Percolation for the Gaussian free field and random interlacements via the cable system

This thesis studies the phase transition associated to the percolation for two different models: level sets of the Gaussian free field and vacant set of random interlacements. The Gaussian free field is a classical model of statistical mechanics, and the study of percolation for its level sets has been initiated by Bricmont and Saleur. Random interlacements is a model recently introduced by Sznitman, and the existence of an infinite component for its vacant set is linked to the existence of a giant component in the vacant set left by a random walk on a torus or a cylinder. These two models have long-range correlations, and compared to the usual independent percolation problem, it is challenging to just prove that the phase transition is non-trivial. We are interested in the existence of a coexistence phase, that is a phase on which the sets of open and closed vertices contain an infinite cluster at the same time. The underlying graph that we consider can be the integer lattice Zd, d > 2, or a more complicated graph, such as a Cayley or a fractal graph with some regularity conditions. One of our main tools is the cable system, a continuous version of the graph, on which one can derive surprisingly explicit results for the percolation of the level sets of the Gaussian free field. This was first noticed by Lupu on Zd, d > 2. Deep results about the existence of a coexistence phase for the discrete Gaussian free field follow from this thorough understanding of the percolative properties on the cable system. A powerful isomorphism between the Gaussian free field and random interlacements, first introduced by Sznitman, leads, in turn, to similar results for random interlacements. In order to understand better the particularities of percolation for the Gaussian free field on the cable system of the integer lattice, we extend and find new results on the cable system of a very general class of graphs using three new independent techniques. For instance, there is no coexistence phase for the level sets of the Gaussian free field on the cable system, and the law of the capacity of a given cluster can be written explicitly

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum