# Random walk with hyperbolic probabilities

## TL;DR

This paper introduces a new type of one-dimensional stochastic process called the random walk with hyperbolic probabilities, which exhibits unique properties blending features of simple and biased random walks, with implications for ergodicity.

## Contribution

It presents the first study of hyperbolic probability-based random walks, revealing their distinctive non-ergodic behavior and providing a geometric interpretation of their transition probabilities.

## Key findings

- The process is not fully ergodic, affecting statistical estimations.
- It combines features of simple and biased random walks.
- Provides a geometric interpretation of transition probabilities.

## Abstract

The random walk with hyperbolic probabilities that we are introducing is an example of stochastic diffusion in a one-dimensional heterogeneous media. Although driven by site-dependent one-step transition probabilities, the process retains some of the features of a simple random walk, shows other traits that one would associate with a biased random walk and, at the same time, presents new properties not related with either of them. In particular, we show how the system is not fully ergodic, as not every statistic can be estimated from a single realization of the process. We further give a geometric interpretation for the origin of these irregular transition probabilities.