Monte Carlo estimation of the number of tatami tilings

TL;DR

This paper introduces a Monte Carlo approach to estimate the number of constrained domino tilings inspired by Japanese tatami mat arrangements, using a novel model and advanced sampling techniques.

Contribution

It develops a new local interaction model on the dual lattice and applies a Monte Carlo method to estimate tiling counts beyond traditional enumeration methods.

Findings

Logarithm of tilings scales linearly with boundary length

Model captures tatami mat placement constraints

Monte Carlo estimates align with known combinatorial results

Abstract

Motivated by the way Japanese tatami mats are placed on the floor, we consider domino tilings with a constraint and estimate the number of such tilings of plane regions. We map the system onto a monomer-dimer model with a novel local interaction on the dual lattice. We use a variant of the Hamiltonian replica exchange Monte Carlo method and the multi-parameter reweighting technique to study the model. The properties of the quantity are studied beyond exact enumeration and combinatorial method. The logarithm of the number of the tilings is linear in the boundary length of the region for all the regions studied.