Low-density series expansions for directed percolation II: The square lattice with a wall

TL;DR

This paper develops a new algorithm to extend low-density series expansions for directed percolation on a square lattice with a wall, providing precise estimates for critical points and exponents, and challenging previous conjectures.

Contribution

A novel algorithm for deriving low-density expansions significantly extends series data for directed percolation near a wall, enabling more accurate critical parameter estimates.

Findings

Critical point estimates match bulk values

Exponent for average cluster length near wall is approximately 1.00014

Results challenge the conjecture that this exponent equals 1

Abstract

A new algorithm for the derivation of low-density expansions has been used to greatly extend the series for moments of the pair-connectedness on the directed square lattice near an impenetrable wall. Analysis of the series yields very accurate estimates for the critical point and exponents. In particular, the estimate for the exponent characterizing the average cluster length near the wall, $\tau_1=1.00014(2)$, appears to exclude the conjecture $\tau_1=1$. The critical point and the exponents $\nu_{\parallel}$ and $\nu_{\perp}$ have the same values as for the bulk problem.