PDF(476 KB)
PDF(476 KB)
PDF(476 KB)
拟顶点可迁图上简单随机游走的切割点
Cutpoints for Simple Random Walks on Quasi-Transitive Graphs
证明了体积增长不低于5次多项式的拟顶点可迁图上的简单随机游走几乎处处有无穷多个切割时，从而有无穷多个切割点.该结论在所论情形下肯定了Benjamini，Gurel-Gurevich和Schramm在文[2011，Cutpoints and resistance ofrandom walk paths，Ann.Probab.，39（3）（3）：1122-1136]中提出的猜想：顶点可迁图上暂留简单随机游走几乎处处有无穷多个切割点.
We prove that a simple random walk on quasi-transitive graphs with the volume growth being at least as fast as a polynomial of degree 5 has a.s. infinitely many cut times, and hence infinitely many cutpoints. This confirms a conjecture raised by Benjamini, Gurel-Gurevich and Schramm[2011, Cutpoints and resistance of random walk paths, Ann. Probab., 39(3):1122-1136] that PATH of a simple random walk on any transient vertex-transitive graph has a.s. infinitely many cutpoints in the corresponding case.
切割点 / 简单随机游走 / 暂留 / 拟顶点可迁图 {{custom_keyword}} /
cutpoint / simple random walk / transience / quasi-transitive graph {{custom_keyword}} /
[1] Alexopoulos G., Random walks on discrete groups of polynomial volume growth, Ann. Probab., 2002, 30(2):723-801.
[2] Benjamini I., Gurel-Gurevich O., Almost sure recurrence of the simple random walk path, 2005, Available at http://arxiv.org/abs/math/0508270.
[3] Benjamini I., Gurel-Gurevich O., Lyons R., Recurrence of random walk traces, Ann. Probab., 2007, 35(2):732-738.
[4] Benjamini I., Gurel-Gurevich O., Schramm O., Cutpoints and resistance of random walk paths, Ann. Probab., 2011, 39(3):1122-1136.
[5] Blachère S., Cut times for random walks on the discrete Heisenberg group, Ann. Inst. H. Poincaré Probab. Statist., 2003, 39(4):621-638.
[6] Csáki E., Földes A., Révész P., On the number of cutpoints of the transient nearest neighbor random walk on the line, J. Theoret. Probab., 2010, 23(2):624-638.
[7] Erdös P., Taylor S. J., Some intersection properties of random walk paths, Acta Math. Acad. Sci. Hungar., 1960, 11, 231-248.
[8] James N., Lyons R., Peres Y., A transient Markov chain with finitely many cutpoints, Probability and Statistics:Essays in Honor of David A. Freedman. Inst. Math. Stat. Collect., 2008, 2:24-29.
[9] James N., Peres Y., Cutpoints and exchangeable events for random walks, Theory Probab. Appl., 1996, 41(4):666-677.
[10] Kochen S., Stone C., A note on the Borel-Cantelli lemma, Illinois J. Math., 1964, 8:248-251.
[11] Lawler G. F., Intersections of Random Walks, Birkhäuser, Boston, 1991.
[12] Lawler G. F., Cut times for simple random walk, Electron. J. Probab., 1996, 1:1-24.
[13] Lyons R., Peres Y., Probability on trees and networks, http://mypage.iu.edu/~rdlyons, Cambridge University Press (Book in preparation), 2016.
[14] Woess W., Random Walks on Infinite Graphs and Groups, Cambridge University Press, Cambridge, 2000.
国家自然科学基金资助项目（11271204，11671216）
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