Hyperscaling for oriented percolation in 1+1 space-time dimensions

TL;DR

This paper proves a hyperscaling relation becomes an equality in 1+1 dimensional oriented percolation, linking critical exponents under certain conditions, using recent crossing probability results.

Contribution

It establishes the exact hyperscaling relation for 1+1 dimensions in oriented percolation, extending understanding of critical phenomena in low-dimensional systems.

Findings

Hyperscaling inequality becomes an equality in 1+1 dimensions.

Relation $ u= ext{eta}+2 ho$ holds under certain conditions.

Uses recent critical crossing probability results.

Abstract

Consider nearest-neighbor oriented percolation in $d+1$ space-time dimensions. Let $\rho,\eta,\nu$ be the critical exponents for the survival probability up to time $t$, the expected number of vertices at time $t$ connected from the space-time origin, and the gyration radius of those vertices, respectively. We prove that the hyperscaling inequality $d\nu\ge\eta+2\rho$, which holds for all $d\ge1$ and is a strict inequality above the upper-critical dimension 4, becomes an equality for $d=1$, i.e., $\nu=\eta+2\rho$, provided existence of at least two among $\rho,\eta,\nu$. The key to the proof is the recent result on the critical box-crossing property by Duminil-Copin, Tassion and Teixeira (2017).