# Random walk in cooling random environment: recurrence versus transience and mixed fluctuations

## TL;DR

This paper studies how resetting the environment at specific times affects the recurrence, transience, and fluctuations of one-dimensional random walks, revealing dependence on the timing of resets.

## Contribution

It extends understanding of RWRE by analyzing how different refreshing times influence recurrence, transience, and fluctuation behaviors in RWCRE.

## Key findings

- Recurrence vs. transience criteria are altered by refreshing times.
- Fluctuation behavior depends delicately on the choice of refreshing times.
- Resetting can significantly modify the correlation structure effects.

## Abstract

This is the third in a series of papers in which we consider one-dimensional Random Walk in Cooling Random Environment (RWCRE). The latter is obtained by starting from one-dimensional Random Walk in Random Environment (RWRE) and resampling the environment along a sequence of deterministic times, called refreshing times. In the present paper we explore two questions for general refreshing times. First, we investigate how the recurrence versus transience criterion known for RWRE changes for RWCRE. Second, we explore the fluctuations for RWCRE when RWRE is either recurrent or satisfies a classical central limit theorem. We show that the answer depends in a delicate way on the choice of the refreshing times. An overarching goal of our paper is to investigate how the behaviour of a random process with a rich correlation structure can be affected by resettings.