# The coin-turning walk and its scaling limit

## TL;DR

This paper studies the scaling limits of a coin-turning random walk under various cooling regimes, revealing a phase transition at a critical scaling where a new recurrent process with Beta marginals emerges.

## Contribution

It introduces a non-classical invariance principle for coin-turning walks, identifying a critical cooling regime and characterizing the limiting process as Beta-distributed and recurrent.

## Key findings

- Invariance principle holds for certain non-classical scalings.
- Critical cooling regime at order const·n^{-1} leads to a Beta-distributed, recurrent process.
- Walk exhibits phase transition in scaling behavior at the critical order.

## Abstract

Let $S$ be the random walk obtained from "coin turning" with some sequence $\{p_n\}_{n\ge 1}$, as introduced in [6]. In this paper we investigate the scaling limits of $S$ in the spirit of the classical Donsker invariance principle, both for the heating and for the cooling dynamics. We prove that an invariance principle, albeit with a non-classical scaling, holds for "not too small" sequences, the order const$\cdot n^{-1}$ (critical cooling regime) being the threshold. At and below this critical order, the scaling behavior is dramatically different from the one above it. The same order is also the critical one for the Weak Law of Large Numbers to hold. In the critical cooling regime, an interesting process emerges: it is a continuous, piecewise linear, recurrent process, for which the one-dimensional marginals are Beta-distributed. We also investigate the recurrence of the walk and its scaling limit, as well as the ergodicity and mixing of the $n$th step of the walk.