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Growth exponent for loop-erased random walk in three dimensions. (English) Zbl 1387.60067
If $$S$$ is the simple random walk on $$Z^d$$ started at the origin and $$\tau_n$$ is its first exit from the ball of radius $$n$$ entered at the origin, how does the random walk path $$S[0,\tau_n]$$ look like?
It is proved that the expected number of cut points is of order $$n^2$$ for $$d>5$$, of order $$n^2(\log n)^{-1/2}$$ for $$d=4$$, and of order $$n^{2-\xi_d}$$ for $$d=2,3$$ for some $$\xi_d$$ which is called the intersection exponent. The value $$\xi_d$$ is known for $$d=2$$, $$\xi_2=5/4$$, and is not known for $$d=3$$.
The author proves that for $$d=3$$, $\lim_{n\to+\infty}\frac{\log E(M_n)}{\log n}=2-\alpha,$ where $$M_n$$ is the length of the loop-erasure of $$S[0,\tau_n]$$, $$\alpha\in [1/3, 1)$$.

##### MSC:
 60G17 Sample path properties 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60D05 Geometric probability and stochastic geometry
##### Keywords:
loop-erased random walk; simple random walk; ergodic theory
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##### References:
 [1] Aaronson, J. (1981). An ergodic theorem with large normalising constants. Israel J. Math.38 182-188. · Zbl 0468.60036 [2] Aaronson, J. and Zweimüller, R. (2014). Limit theory for some positive stationary processes with infinite mean. Ann. Inst. Henri Poincaré Probab. Stat.50 256-284. · Zbl 1291.60067 [3] Barlow, M. T. and Masson, R. (2010). Exponential tail bounds for loop-erased random walk in two dimensions. Ann. Probab.38 2379-2417. · Zbl 1207.60035 [4] Benjamini, I., Gurel-Gurevich, O. and Lyons, R. (2007). Recurrence of random walk traces. Ann. Probab.35 732-738. · Zbl 1118.60059 [5] Benjamini, I., Gurel-Gurevich, O. and Schramm, O. (2011). Cutpoints and resistance of random walk paths. Ann. Probab.39 1122-1136. · Zbl 1223.60012 [6] Bolthausen, E., Sznitman, A.-S. and Zeitouni, O. (2003). Cut points and diffusive random walks in random environment. Ann. Inst. Henri Poincaré Probab. Stat.39 527-555. · Zbl 1016.60094 [7] Croydon, D. A. (2009). Random walk on the range of random walk. J. Stat. Phys.136 349-372. · Zbl 1184.60011 [8] Damron, M. and Sapozhnikov, A. (2011). Outlets of 2D invasion percolation and multiple-armed incipient infinite clusters. Probab. Theory Related Fields 150 257-294. · Zbl 1225.82030 [9] Doyle, P. G. and Snell, J. L. (1984). Random Walks and Electric Networks. Carus Mathematical Monographs 22. Mathematical Association of America, Washington, DC. · Zbl 0583.60065 [10] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Series in Statistical and Probabilistic Mathematics 31. Cambridge Univ. Press, Cambridge. [11] Guttmann, J. and Bursill, R. J. (1990). Critical exponents for the loop erased self-avoiding walk by Monte Carlo methods. J. Stat. Phys.59 1-9. [12] James, N. and Peres, Y. (1996). Cutpoints and exchangeable events for random walks. Teor. Veroyatn. Primen.41 854-868. · Zbl 0896.60035 [13] Kenyon, R. (2000). The asymptotic determinant of the discrete Laplacian. Acta Math.185 239-286. · Zbl 0982.05013 [14] Kozma, G. (2007). The scaling limit of loop-erased random walk in three dimensions. Acta Math.199 29-152. · Zbl 1144.60060 [15] Lawler, G. F. (1980). A self-avoiding random walk. Duke Math. J.47 655-693. · Zbl 0445.60058 [16] Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser, Inc., Boston, MA. · Zbl 1228.60004 [17] Lawler, G. F. (1992). Escape probabilities for slowly recurrent sets. Probab. Theory Related Fields 94 91-117. · Zbl 0767.60062 [18] Lawler, G. F. (1996). Cut times for simple random walk. Electron. J. Probab.1 no. 13, 24 pp. (electronic). · Zbl 0888.60059 [19] Lawler, G. F. (1996). Hausdorff dimension of cut points for Brownian motion. Electron. J. Probab.1 paper no. 2. · Zbl 0891.60078 [20] Lawler, G. F. (1999). Loop-erased random walk. In Perplexing Problems in Probability: Festschrift in Honor of Harry Kesten (M. Bramson and R. T. Durrett, eds.). Progress in Probability 44 197-217. Birkhauser Boston, Boston, MA. · Zbl 0947.60055 [21] Lawler, G. F. (2005). Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs 114. Amer. Math. Soc., Providence, RI. · Zbl 1074.60002 [22] Lawler, G. F. (2014). The probability that planar loop-erased random walk uses a given edge. Electron. Commun. Probab.19 no. 51, 13. · Zbl 1303.82027 [23] Lawler, G. F. and Limic, V. (2010). Random Walk: A Modern Introduction. Cambridge Univ. Press. · Zbl 1210.60002 [24] Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. II. Plane exponents. Acta Math.187 275-308. · Zbl 0993.60083 [25] Lawler, G. F., Schramm, O. and Werner, W. (2004). Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab.32 939-995. · Zbl 1126.82011 [26] Lawler, G. F. and Vermesi, B. (2010). Fast convergence to an invariant measure for non-intersecting 3-dimensional Brownian paths. Preprint, available at http://arxiv.org/abs/1008.4830. · Zbl 1290.60080 [27] Masson, R. (2009). The growth exponent for planar loop-erased random walk. Electron. J. Probab.14 1012-1073. · Zbl 1191.60061 [28] Moerters, P. and Peres, Y. (2010). Brownian Motion. Cambridge Univ. Press. [29] Pemantle, R. (1991). Choosing a spanning tree for the integer lattice uniformly. Ann. Probab.19 1559-1574. · Zbl 0758.60010 [30] Sapozhnikov, A. and Shiraishi, D. On Brownian motion, simple paths, and loops. Available at http://arxiv.org/abs/1512.04864. · Zbl 1404.60062 [31] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math.118 221-288. · Zbl 0968.60093 [32] Shiraishi, D. Random walk on non-intersecting two-sided random walk trace is subdiffusive in low dimensions. Trans. Amer. Math. Soc. To appear. · Zbl 1388.82014 [33] Shiraishi, D. (2010). Heat kernel for random walk trace on $$\mathbb{Z}^{3}$$ and $$\mathbb{Z}^{4}$$. Ann. Inst. Henri Poincaré Probab. Stat.46 1001-1024. · Zbl 1208.82048 [34] Shiraishi, D. (2012). Exact value of the resistance exponent for four dimensional random walk trace. Probab. Theory Related Fields 153 191-232. · Zbl 1246.82044 [35] Shiraishi, D. (2012). Two-sided random walks conditioned to have no intersections. Electron. J. Probab.17 no. 18, 24 pp. · Zbl 1244.05209 [36] Wilson, D. B. (1996). Generating random spanning trees more quickly than the cover time. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996) 296-303. ACM, New York. · Zbl 0946.60070 [37] Wilson, D.
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