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

[1]  A. Cayley.  A theorem on trees , 2009 .

[2]  Matthias Schulz,et al.  Addition of Recurrent Configurations in Chip Firing Games: Finding Minimal Recurrent Configurations with Markov Chains , 2010, ACRI.

[3]  David B. Wilson,et al.  Chip-Firing and Rotor-Routing on Directed Graphs , 2008, 0801.3306.

[4]  D. Dhar,et al.  Algebraic aspects of Abelian sandpile models , 1995 .

[5]  Norman Biggs,et al.  Chip-Firing and the Critical Group of a Graph , 1999 .

[6]  Matthew Baker,et al.  Chip-firing games, potential theory on graphs, and spanning trees , 2011, J. Comb. Theory, Ser. A.

[7]  Thomas H. Parker,et al.  What is π , 1991 .

[8]  Peter Mark Kayll,et al.  A chip-firing variation and a new proof of Cayley's Formula , 2013, Discret. Math. Theor. Comput. Sci..

[9]  Gábor Tardos,et al.  Polynomial Bound for a Chip Firing Game on Graphs , 1988, SIAM J. Discret. Math..

[10]  Stassinopoulos,et al.  Democratic reinforcement: A principle for brain function. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  Lionel Levine,et al.  Abelian networks II: halting on all inputs , 2014, ArXiv.

[12]  Lionel Levine,et al.  Abelian networks III: The critical group , 2014, ArXiv.

[13]  Dhar,et al.  Self-organized critical state of sandpile automaton models. , 1990, Physical review letters.

[14]  Frank Harary,et al.  Graph Theory , 2016 .

[15]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[16]  Paczuski,et al.  Emergent traffic jams. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  Casey A. Volino,et al.  A First Course in Stochastic Models , 2005, Technometrics.

[18]  Investigations of a chip-firing game , 2005 .

[19]  Caroline J. Klivans,et al.  Chip-firing and energy minimization on M-matrices , 2015, J. Comb. Theory, Ser. A.

[20]  Thi Ha Duong Phan,et al.  Lattices generated by Chip Firing Game models: Criteria and recognition algorithms , 2012, Eur. J. Comb..

[21]  Prasad Tetali,et al.  G-parking functions, acyclic orientations and spanning trees , 2008, Discret. Math..

[22]  R. Meester,et al.  Existence and uniqueness of the stationary measure in the continuous Abelian sandpile , 2009, 0911.3782.

[23]  László Lovász,et al.  Chip-firing Games on Graphs , 1991, Eur. J. Comb..

[24]  László Lovász,et al.  Chip-Firing Games on Directed Graphs , 1992 .

[25]  Jan van den Heuvel,et al.  Algorithmic Aspects of a Chip-Firing Game , 2001, Combinatorics, Probability and Computing.

[26]  Jörg Lingens,et al.  The Growth Model , 2004 .

[27]  Lionel Levine,et al.  Abelian Networks I. Foundations and Examples , 2013, SIAM J. Discret. Math..

[28]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[29]  David Bruce Wilson,et al.  How to Get a Perfectly Random Sample from a Generic Markov Chain and Generate a Random Spanning Tree of a Directed Graph , 1998, J. Algorithms.

[30]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[31]  P. Bak,et al.  Earthquakes as a self‐organized critical phenomenon , 1989 .

[32]  Criel Merino,et al.  The chip-firing game , 2005, Discret. Math..

[33]  P. Kayll,et al.  Combinatorial Proof of an Abel-type Identity ∗ , 2009 .