Site-percolation transition of run-and-tumble particles

TL;DR

This paper investigates the percolation transition of run-and-tumble particles on a 2D lattice, revealing a re-entrant phase transition with critical exponents related to Ising model universality.

Contribution

It introduces a detailed analysis of the site-percolation transition in RTPs, showing continuous variation of critical exponents and invariance of a universal scaling function.

Findings

Identification of a phase-separated state at low tumble rates

Observation of a re-entrant percolation transition with varying parameters

Establishment of a connection between motility-induced phase separation and percolation critical exponents

Abstract

We study percolation transition of run and tumble particles (RTPs) on a two dimensional square lattice. RTPs in these models run to the nearest neighbour along their internal orientation with unit rate, and to other nearest neighbours with rates $p$. In addition, they tumble to change their internal orientation with rate $\omega$. We show that for small tumble rates, RTP-clusters created by joining occupied nearest neighbours irrespective of their orientation form a phase separated state when the rate of positional diffusion $p$ crosses a threshold; with further increase of $p$ the clusters disintegrate and another transition to a mixed phase occurs. The critical exponents of this re-entrant site-percolation transition of RTPs vary continuously along the critical line in the $\omega$-$p$ plane, but a scaling function remains invariant. This function is identical to the corresponding universal scaling function of percolation transition observed in the Ising model. We also show that the critical exponents of the underlying motility induced phase separation transition are related to corresponding percolation-critical-exponents by constant multiplicative factors known from the correspondence of magnetic and percolation critical exponents of the Ising model.