Nikita Titov — University of Trieste #
From Laplacian-to-Adjacency matrix for continuous spins on graphs  #
The study of spins and particles on graphs has applications across many areas, from time dynamics on networks to combinatorial optimization. In this talk, I will discuss the large n limit of the O(n) model on general graphs. The absence of translational invariance leads to an infinite set of saddle point constraints in the thermodynamic limit. I will show that the free energy at low and high temperatures T is controlled by two central graph-theoretic objects: the Laplacian matrix at low T and the Adjacency matrix at high T. Their interplay will be illustrated across several classes of graphs. On trees, on can obtain an exact solution in which the Lagrange multipliers depend only on the local coordination numbers. I will highlight the physical consequences using the example of a Y-junction. For graphs allowing a phase transition I will show that the singular part of the free energy is governed by the Laplacian spectrum, whereas the full free energy coincides with it only in the zero-temperature limit. I will conclude by presenting analogous results for the quantum version of the model.