Anomalous dynamical scaling determines universal critical singularities

Abstract

Anomalous diffusion phenomena occur on length scales spanning from intracellular to astrophysical ranges. A specific form of decay at large argument of the probability density function of rescaled displacement (scaling function) is derived and shown to imply universal singularities in the normalized cumulant generator. Exact calculations for continuous time random walks provide paradigmatic examples connected with singularities of second order phase transitions. In the biased case scaling is restricted to displacements in the drift direction and singularities have no equilibrium analogue.

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• Generalized arcsine laws for a sluggish random walker with subdiffusive growth
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• Generalization of the Central Limit Theorem to Critical Systems: Revisiting Perturbation Theory
• Universal and non-universal large deviations in critical systems
• Critical Probability Distributions of the order parameter at two loops I: Ising universality class
• Systematic analysis of critical exponents in continuous dynamical phase transitions of weak noise theories
• Proxitaxis: an adaptive search strategy based on proximity and stochastic resetting
• Universal and non-universal signatures in the scaling functions of critical variables
• Understanding random-walk dynamical phase coexistence through waiting times
• Critical Probability Distributions of the order parameter at two loops II: $O(n)$ universality class