Exponential Bounds and Analyticity for the Tree Builder Random Walk

TL;DR

This paper studies a class of random walks called Tree Builder Random Walks, proving exponential tail bounds for renewal times and analyticity of the speed in certain cases, using complex analysis techniques.

Contribution

It establishes exponential tail bounds for the first renewal time and proves the speed's analyticity as a function of parameters in specific scenarios.

Findings

First renewal time has exponential tail.

Empirical speed is exponentially bounded.

Speed is analytic in parameter p when adding one vertex with probability p.

Abstract

In this work we investigate a class of random walks that interacts with its environment called Tree Builder Random Walk (TBRW). In our settings, at each step, the walker adds a random number of vertices to its position sampled according to a distribution $Q$. Previous works showed that the walker is ballistic with a well-defined speed, and that the TBRW admits a renewal structure, meaning that the process can be split into i.i.d epochs. We show that the first renewal time has exponential tail. Moreover, we show two consequences of the light tail of the first renewal time: an exponential upper bound for the empirical speed of the walker, and, for the case in which the walker adds only one vertex with probability $p$, we show that the limiting speed is an analytic function of the parameter $p$. In some of our proofs, we apply techniques from bond percolation, which consist of extending probabilities to the complex numbers and using the Weierstrass $M$-test.