On the meeting of random walks on random DFA

TL;DR

This paper analyzes the meeting time of two synchronized random walks on a random DFA, showing it is typically short and tightly bounded, with implications for computational learning.

Contribution

It establishes tight bounds on the meeting time of random walks on random DFAs, connecting probabilistic analysis with learning theory conjectures.

Findings

Meeting time is stochastically dominated by a geometric distribution with rate ~1/n.

Upper bounds on meeting time are tight with high probability.

Results have implications for understanding random processes in automata and learning theory.

Abstract

We consider two random walks evolving synchronously on a random out-regular graph of $n$ vertices with bounded out-degree $r\ge 2$, also known as a random Deterministic Finite Automaton (DFA). We show that, with high probability with respect to the generation of the graph, the meeting time of the two walks is stochastically dominated by a geometric random variable of rate $(1+o(1))n^{-1}$, uniformly over their starting locations. Further, we prove that this upper bound is typically tight, i.e., it is also a lower bound when the locations of the two walks are selected uniformly at random. Our work takes inspiration from a recent conjecture by Fish and Reyzin in the context of computational learning, the connection with which is discussed.