A percolation model with continuously varying exponents

TL;DR

This paper introduces a percolation model on the diamond hierarchical lattice where critical exponents vary continuously with erasing probability, revealing new universality classes and phase transition behaviors.

Contribution

It provides analytical and numerical analysis of a modified percolation model with continuously varying exponents, connecting it to Potts models with long-range interactions.

Findings

Critical exponents $ u$ and $eta$ vary continuously with erasing probability.

The model can exhibit $ u= ext{infinity}$, indicating a distinct phase transition.

Percolation transition remains continuous with $eta$ approaching zero.

Abstract

This work analyzes a percolation model on the diamond hierarchical lattice (DHL), where the percolation transition is retarded by the inclusion of a probability of erasing specific connected structures. It has been inspired by the recent interest on the existence of other universality classes of percolation models. The exact scale invariance and renormalization properties of DHL leads to recurrence maps, from which analytical expressions for the critical exponents and precise numerical results in the limit of very large lattices can be derived. The critical exponents $\nu$ and $\beta$ of the investigated model vary continuously as the erasing probability changes. An adequate choice of the erasing probability leads to the result $\nu=\infty$, like in some phase transitions involving vortex formation. The percolation transition is continuous, with $\beta>0$, but $\beta$ can be as small as desired. The modified percolation model turns out to be equivalent to the $Q\rightarrow1$ limit of a Potts model with specific long range interactions on the same lattice.