Exponents for Hamiltonian paths on random bicubic maps and KPZ

TL;DR

This paper investigates the critical exponents of Hamiltonian paths on random bicubic maps, compares empirical estimates with KPZ predictions, and proposes modifications to reconcile discrepancies, enhancing understanding of these combinatorial structures.

Contribution

It introduces a modified application of KPZ relations to better match empirical exponents for Hamiltonian paths and fully packed loops on random bicubic maps.

Findings

Naive KPZ application does not match empirical exponents

A simple modification improves the match between theory and data

Modified KPZ relations can reproduce known exponents for certain models

Abstract

We evaluate the configuration exponents of various ensembles of Hamiltonian paths drawn on random planar bicubic maps. These exponents are estimated from the extrapolations of exact enumeration results for finite sizes and compared with their theoretical predictions based on the KPZ relations, as applied to their regular counterpart on the honeycomb lattice. We show that a naive use of these relations does not reproduce the measured exponents but that a simple modification in their application may possibly correct the observed discrepancy. We show that a similar modification is required to reproduce via the KPZ formulas some exactly known exponents for the problem of unweighted fully packed loops on random planar bicubic maps.