Abstract
We describe two possible regimes (dynamic phases) of the Euler walk on a Cayley tree: a condensed phase and a low-density phase. In the condensed phase the area of visited sites grows as a compact domain. In the low-density phase the proportion of visited sites decreases rapidly from one generation of the tree to the next. We describe in detail returns of the walker to the root and growth of the domain of visited sites in the condensed phase. We also investigate the critical behaviour of the model on the line separating the two regimes.
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References
P. Alstrø m, Mean-field exponents for self-organized critical phenomena. Phys. Rev. A 38:4905–4906 (1988).
P. Bak, C. Tang, and K. Wiesenfeld, Self-organized criticallity. Phys. Rev. A 38:364–374 (1988).
E. Buffet, A. Patrick, and J. Pule, Directed polymers on trees: A martingale approach. J. Phys. A: Math. Gen. 26:1823–1834 (1993).
D. Dhar and S. N. Majumdar, Abelian sandpile model on the Bethe lattice. J. Phys. A: Math. Gen. 23:4333–4350 (1990).
W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1 (John Wiley, New York, 1970).
T. E. Harris, The Theory of Branching Processes (Springer-Verlag, Berlin, 1963).
R. Otter, The multiplicative process. Ann. Math. Stat. 20:206–224 (1949).
Vl. V. Papoyan and V. B. Priezzhev, Private discussion (2006).
A. M. Povolotsky, V. B. Priezzhev, and R. R. Shcherbakov, Dynamics of Eulerian Walkers. Phys. Rev. E 58:5449–5454 (1998).
V. B. Priezzhev, D. Dhar, A. Dhar, and S. Krishnamurthy, Eulerian Walkers as a model of self-organized criticality. Phys. Rev. Lett. 77:5079–5082 (1996).
A. N. Shiryaev, Probability (Springer-Verlag, Berlin, 1998).
S. Zapperi, K. B. Lauritsen, and H. E. Stanley, Self-organized branching process: Mean-field theory for avalanches. Phys. Rev. Lett. 75:4071–4074 (1995).
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Patrick, A.E. Euler Walk on a Cayley Tree. J Stat Phys 127, 629–653 (2007). https://doi.org/10.1007/s10955-007-9281-5
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DOI: https://doi.org/10.1007/s10955-007-9281-5