A chip-firing variation and a Markov chain with uniform stationary distribution

We continue our study of burn-off chip-firing games on graphs initiated in [Discrete Math. Theor. Comput. Sci. 15 (2013), no. 1, 121–132]. Here we introduce randomness by choosing each successive seed uniformly from among all possible nodes. The resulting stochastic process—a Markov chain (Xn)n≥0 with state space the set R of relaxed legal chip configurations C : V → N on a connected graph G = (V,E)—has the property that, with high probability, each state appears equiproportionally in a long sequence of burn-off games. This follows from our main result that (Xn) has a doubly stochastic transition matrix. As tools supporting our main proofs, we establish several properties of the chip addition operator E(·)(·) : V ×R → R that may be of independent interest. For example, if V = {v1, . . . , vn}, then C ∈ R if and only if C is fixed by the composition Ev1 ◦ · · · ◦ Evn ; this property eventually yields the irreducibility of (Xn). ∗ Part of this work appears in the author’s PhD dissertation [29] † This work was partially supported by a grant from the Simons Foundation (#279367 to Mark Kayll). D. PERKINS AND P.M. KAYLL /AUSTRALAS. J. COMBIN. 68 (3) (2017), 330–345 331