Speaker
Description
While universality on regular lattices is controlled by the Euclidean dimension, its fate on inhomogeneous structures has long remained an open question, with early work identifying the spectral dimension as the relevant parameter. In this talk, I will consider simple random graphs where long-range bonds are drawn from a Bernoulli distribution, so that the same bonds simultaneously generate superdiffusive transport and correlated quenched disorder — a situation beyond the conventional Harris and Weinrib–Halperin framework, where disorder perturbs a pre-existing clean theory. For the self-avoiding walk on such graphs, large-scale Monte Carlo simulations and a Gaussian-truncated field theory show that the long-range kinetic operator dominates under coarse-graining, restoring the clean superdiffusive Lévy universality class, yet through an irrelevance mechanism qualitatively different from the Weinrib–Halperin scenario. In contrast, in the single-particle spectral problem the same geometric randomness generates quantum-chaotic level statistics and drives a localization transition beyond the reach of Gaussian field theory, signalling the infrared relevance of higher Bernoulli cumulants. These results identify graph disorder as a distinct class of randomness, whose non-Gaussian statistics may give rise to novel universality classes.