Excursion Processes Associated with Elliptic Combinatorics

TL;DR

This paper introduces elliptic excursion processes linked to elliptic combinatorics, revealing nontrivial curved trajectories and asymptotic behaviors, thus offering new models for non-equilibrium statistical mechanics.

Contribution

It develops a family of elliptic excursion processes with inhomogeneous weights, connecting elliptic combinatorics to stochastic processes and statistical mechanics.

Findings

Maximum likelihood trajectories are non-straight and curved.

Asymptotic laws show emergence of nontrivial large-scale trajectories.

Elliptic weight-functions produce complex, curved paths in space-time.

Abstract

Researching elliptic analogues for equalities and formulas is a new trend in enumerative combinatorics which has followed the previous trend of studying $q$-analogues. Recently Schlosser proposed a lattice path model in the square lattice with a family of totally elliptic weight-functions including several complex parameters and discussed an elliptic extension of the binomial theorem. In the present paper, we introduce a family of discrete-time excursion processes on $\mathbb{Z}$ starting from the origin and returning to the origin in a given time duration $2T$ associated with Schlosser's elliptic combinatorics. The processes are inhomogeneous both in space and time and hence expected to provide new models in non-equilibrium statistical mechanics. By numerical calculation we show that the maximum likelihood trajectories on the spatio-temporal plane of the elliptic excursion processes and of their reduced trigonometric versions are not straight lines in general but are nontrivially curved depending on parameters. We analyze asymptotic probability laws in the long-term limit $T \to \infty$ for a simplified trigonometric version of excursion process. Emergence of nontrivial curves of trajectories in a large scale of space and time from the elementary elliptic weight-functions exhibits a new aspect of elliptic combinatorics.