Vacant Sets and Vacant Nets: Component Structures Induced by a Random Walk

Given a discrete random walk on a finite graph G, the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step t. Let Γ(t) be the subgraph of G induced by the vacant set of the walk at step t. Similarly, let Γ̂(t) be the subgraph of G induced by the edges of the vacant net. For random r-regular graphs Gr, it was previously established that for a simple random walk, the graph Γ(t) of the vacant set undergoes a phase transition in the sense of the phase transition on Erdős-Renyi graphs Gn,p. Thus, for r ≥ 3 there is an explicit value t∗ = t∗(r) of the walk, such that for t ≤ (1− )t∗, Γ(t) has a unique giant component, plus components of size O(log n), whereas for t ≥ (1 + )t∗ all the components of Γ(t) are of size O(log n). In this paper we establish the threshold value t̂ for a phase transition in the graph Γ̂(t) of the vacant net of a simple random walk on a random r-regular graph,. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a non-backtracking random walk, and, for r even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random r-regular graphs. The main findings are the following: As r increases the threshold for the vacant set converges to n log r in all three walks. For the vacant net, the threshold converges to rn/2 log n for both the simple random walk and non-backtracking random walk. When r ≥ 4 is even, the threshold for the vacant net of the unvisited edge process converges to rn/2, which is also the vertex cover time of the process. ∗Department of Informatics, King’s College, University of London, London WC2R 2LS, UK. Research supported in part by EPSRC grants EP/J006300/1 and EP/M005038/1. †Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA 15213, USA. Research supported in part by NSF grant DMS0753472.