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The scaling limits of planar LERW in finitely connected domains. (English) Zbl 1153.60057
The loop-erased random walk (LERW) is obtained by removing loops from a simple random walk on a graph that is stopped at some hitting time [G.F. Lawler, Intersections of random walks. Probability and Its Applications. Boston: Birkhäuser (1996; Zbl 0925.60078)]. In this paper, the loop erasures of conditional random walks are considered. They have properties rather similar to the LERW, so they are called LERWs as well. In [O. Schramm, Isr. J. Math. 118, 221–288 (2000; Zbl 0968.60093)], the stochastic Loewner evolution (SLE) was introduced, i.e., a family of random growth processes of closed fractal subsets in simply connected plane domains. The evolution was described by the classical Loewner equation with driving term being \(\sqrt{k}\) times that a standard linear Brownian motion for some \(k\geq 0\). In this paper, a family of SLE-type processes is defined in finitely connected domains, called continuous LERW, using the usual chordal Loewner equation with the driving function \(\sqrt{2}B(t)+S(t)\), where \(B(t)\) is a standard linear Brownian motion and the drift term \(S(t)\) is such that the continuous LERW satisfies conformal invariance and preserves a family of local martingales generated by generalized Poisson kernels. The local martingales are similar to the discrete ones preserved by the corresponding discrete LERW on the discrete approximation of that domain. This similarity is employed to prove the convergence of discrete LERW to continuous LERW. A continuous LERW describes a random curve in a finitely connected domain starting from a prime end and ends at a certain target set which could be an interior point or a prime end or a side arc.
In section 2, the notions of the loop-erased random walk, finitely connected domains, positive harmonic function, hulls and Loewner chains are discussed rather widely for the needs of the paper. Section 3 deals with three kinds of continuous LERWs. The chordal Loewner equation, continuous LERWs aiming at the interior points, conformal invariance and continuous LERWs with other kinds of targets are considered. The continuous or discrete LERWs and the similarity between them are discussed in Section 4. In Section 5, the proof of existence and uniqueness of the solution of the equation describing a continuous LERW is presented. The Caratheodory topology is applied to define the convergence of plain domains. The metric on the space of hulls in the upper half plane is defined so that the set of hulls, contained in a fixed hull, is compact. This compactness is frequently used in the paper. The LERWs whose targets are the interior points are considered in Section 6. Besides, it is proved that the driving function of the discrete LERW converges to that of the continuous LERW. In Section 7, some regular properties of the discrete LERW curve are applied to get the local coupling of the LERW curve and the continuous LERW trace so that the two curves are close to each other before one of them leaves a hull bounded by a crosscut. Finally, all local couplings are applied to get the statement that with probability close to \(1\), the two curves are uniformly close to each other. In the last section, the convergence of LERWs of two other kinds is considered. Similar results are obtained.
The volume of the paper is 63 pages and the list of references contains 21 positions.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G17 Sample path properties
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
60G50 Sums of independent random variables; random walks
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References:
[1] Lars, V. A. (1973). Conformal Invariants: Topics in Geometric Function Theory . McGraw-Hill Book Co., New York. · Zbl 0272.30012
[2] Bauer, M., Bernard, D. and Houdayer, J. (2005). Dipolar stochastic Loewner evolutions. J. Stat. Mech. P03001.
[3] Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201
[4] Lawler, G. F. (1991). Intersection of Random Walks . Birkhäuser, Boston. · Zbl 0735.60071
[5] Lawler, G. F. (2005). Conformally Invariant Processes in the Plane . Amer. Math. Soc., Providence, RI. · Zbl 1074.60002
[6] Lawler, G. F., Schramm, O. and Werner, W. (2001). Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187 237-273. · Zbl 1005.60097
[7] 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
[8] Lawler, G. F., Schramm, O. and Werner, W. (2002). Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. H. Poincaré Probab. Statist. 38 109-123. · Zbl 1006.60075
[9] Lawler, G. F., Schramm, O. and Werner, W. (2003). Conformal restriction: The chordal case. J. Amer. Math. Soc. 16 917-955. · Zbl 1030.60096
[10] 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
[11] Oksendal, B. (1995). Stochastic Differential Equations: An Introduction with Applications , 4th ed. Springer, Berlin.
[12] Pommerenke, C. (1975). Univalent Functions . Vandenhoeck & Ruprecht, Göttingen. · Zbl 0298.30014
[13] Pommerenke, C. (1991). Boundary Behaviour of Conformal Maps . Springer, Berlin. · Zbl 0762.30001
[14] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion . Springer, Berlin. · Zbl 0731.60002
[15] Rohde, S. and Schramm, O. (2005). Basic properties of SLE. Ann. Math. (2) 161 883-924. · Zbl 1081.60069
[16] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221-288. · Zbl 0968.60093
[17] Schramm, O. and Sheffield, S. (2005). The harmonic explorer and its convergence to SLE 4 . Ann. Probab. 33 2127-2148. · Zbl 1095.60007
[18] Smirnov, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér.I Math. 333 239-244. · Zbl 0985.60090
[19] Werner, W. (2004). Random planar curves and Schramm-Loewner evolutions. Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1840 107-195. Springer, Berlin. · Zbl 1057.60078
[20] Zhan, D. (2004). Stochastic Loewner evolution in doubly connected domains. Probab. Theory Related Fields 129 340-380. · Zbl 1054.60104
[21] Zhan, D. (2004). Random Loewner chains in Riemann surfaces. Ph.D. dissertation, Caltech.
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