Combinatorial expressions of the solutions to initial value problems of the discrete and ultradiscrete Toda molecules

TL;DR

This paper derives combinatorial solutions for initial value problems of discrete and ultradiscrete Toda molecules, using non-intersecting paths and shortest path algorithms, connecting combinatorics with integrable systems.

Contribution

It introduces a novel combinatorial framework for solving Toda molecule initial value problems, including ultradiscrete cases, via path counting and graph algorithms.

Findings

Subtraction-free expressions for solutions using non-intersecting paths

Ultradiscrete solutions obtained through ultradiscretization

Exact solutions expressed as shortest paths on specific graphs

Abstract

Combinatorial expressions are presented to the solutions to initial value problems of the discrete and ultradiscrete Toda molecules. For the discrete Toda molecule, a subtraction-free expression of the solution is derived in terms of non-intersecting paths, for which two results in combinatorics, Flajolet's interpretation of continued fractions and Gessel--Viennot's lemma on determinants, are applied. By ultradiscretizing the subtraction-free expression, the solution to the ultradiscrete Toda molecule is obtained. It is finally shown that the initial value problem of the ultradiscrete Toda molecule is exactly solved in terms of shortest paths on a specific graph.