# Zero Range Process and Multi-Dimensional Random Walks

## TL;DR

This paper explores the connection between zero range processes, non-Hermitian Hamiltonians, and multi-dimensional lattice walks, providing algebraic solutions and expressing probabilities through symmetric functions.

## Contribution

It introduces a novel interpretation of conditional probabilities as generating functions for multi-dimensional lattice walks bounded by hyperplanes.

## Key findings

- Conditional probabilities are expressed as generating functions for walks.
- Conditional probabilities and walk counts are formulated via symmetric functions.
- The approach links non-equilibrium statistical mechanics with combinatorial lattice walk analysis.

## Abstract

The special limit of the totally asymmetric zero range process of the low-dimensional non-equilibrium statistical mechanics described by the non-Hermitian Hamiltonian is considered. The calculation of the conditional probabilities of the model are based on the algebraic Bethe ansatz approach. We demonstrate that the conditional probabilities may be considered as the generating functions of the random multi-dimensional lattice walks bounded by a hyperplane. This type of walks we call the walks over the multi-dimensional simplicial lattices. The answers for the conditional probability and for the number of random walks in the multi-dimensional simplicial lattice are expressed through the symmetric functions.