Estimates of Random Walk Exit Probabilities and Application to Loop-Erased Random Walk

Schramm-Loewner Evolutions (SLEs) describe a one-parameter family of growth processes in the plane that have particular conformal invariance properties. For instance, SLE can define simple random curves in a simply connected domain. In this paper we are interested in questions pertaining to the definition of several SLEs in a domain (i.e. several random curves). In particular, one derives infinitesimal commutation conditions, discuss some elementary solutions, study integrability conditions following from commutation and show how to lift these infinitesimal relations to global relations in simple cases. The situation in multiply-connected domains is also discussed.For plane critical models of statistical physics, such as percolation or the Ising model, the general line of thinking of Conformal Field Theory leads to expect the existence of a non-degenerate scaling limit that satisfies conformal invariance properties. Though, it is not quite clear how to define this scaling limit and what conformal invariance exactly means.One way to proceed is to consider a model in a, say, bounded (plane) simply-connected domain with Jordan boundary, and to set boundary conditions so as to force the existence of a macroscopic interface connecting two marked points on the boundary. In this set-up, Schramm has shown that the possible scaling limits satifying conformal invariance along with a "domain Markov" property are classified by a single positive parameter κ > 0, in the seminal article [19]. This defines the family of Schramm-Loewner Evolutions (SLEs), that are probability measures supported on non-self-traversing curves connecting two marked boundary points in a simply-connected domain.Consider the following situation for critical site percolation on the triangular lattice: a portion of the triangular lattice with mesh ε approximates a fixed simply connected domain D with two points x and y marked on the boundary. The boundary arc (xy) is set to blue and (yx) is set to yellow; sites are blue or yellow with probability 1/2. Then the interface between blue sites connected to (xy) and yellow sites connected to (yx) is a non-self traversing curve from x to y. In this set-up, Smirnov has proved that the interface converges to SLE 6 , as conjectured earlier by Schramm ([21, 4]).For discrete models such as percolation or the Ising model, the full information can be encoded as a collection of contours (interfaces between blue any yellow, + and − spins, . . . ). Hence it is quite natural to consider scaling limits as collection of countours, as in [1,4]. Comparing the ideas of isolating one macroscopic interface by setting appropriate boundary conditions (following Schramm), and considering the scaling limit as a collection of contours, one is led to the problem of describing the joint law of a finite number of macroscopic interfaces created by appropriate boundary conditions. For instance, for percolation, consider a simply connected domain with 2n marked points on the boundary, the 2n boundary arcs being alternatively blue ...

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“…This situation is discussed in details in [10]; when κ = 2, this is directly connected to Fomin's formulae [11,12].…”

Section: Localitymentioning

confidence: 99%

Commutation relations for Schramm‐Loewner evolutions

Dubédat

2007

Comm Pure Appl Math

123

266

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“…Kenyon [19] used this relation to show that the average number of steps in a LERW on an N ×N box is N 5/4 . The process is also a determinantal process and this combined with conformal invariance of Brownian motion can establish conformal invariance for some quantities without discussing SLE [14,20]. In [26], LSW established that the scaling limit of looperased random walk is SLE 2 and also the scaling limit of uniform spanning trees (appropriately defined) gives SLE 8 .…”

Section: Brownian Pathsmentioning

confidence: 99%

Conformal invariance and $2D$ statistical physics

Lawler

2008

Bull. Amer. Math. Soc.

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Abstract. A number of two-dimensional models in statistical physics are conjectured to have scaling limits at criticality that are in some sense conformally invariant. In the last ten years, the rigorous understanding of such limits has increased significantly. I give an introduction to the models and one of the major new mathematical structures, the Schramm-Loewner Evolution (SLE). Critical phenomenaCritical phenomena in statistical physics refers to the study of systems at or near the point at which a phase transition occurs. There are many models of such phenomena. We will discuss some discrete equilibrium models that are defined on a lattice. These are measures on paths or configurations where configurations are weighted by their energy E with a preference for paths of smaller energy. These measures depend on at least one parameter. A standard parameter in physics is β = c/T , where c is a fixed constant, T stands for temperature, and the measure given to a configuration is e −βE . Phase transitions occur at critical values of the temperature corresponding to, e.g., the transition from a gaseous to a liquid state. We use β for the parameter although for understanding the literature it is useful to know that large values of β correspond to "low temperature" and small values of β correspond to "high temperature". Small β (high temperature) systems have weaker correlations than large β (low temperature) systems. In a number of models, there is a critical β c such that qualitatively the system has three regimes β < β c (high temperature), β = β c (critical), and β > β c (low temperature).A standard procedure is to define a model on a finite subset of a lattice, and then let the lattice size grow. Equivalently, one can consider a bounded region and consider finer and finer lattices inside the region. In either case, one would like to know the scaling or continuum limit of the system. For many models, it can be difficult to even describe what kind of object one has in the limit, and then it can be much more difficult to prove such a limit exists.The behavior of the models described iin this paper varies significantly in different dimensions. In two dimensions, Belavin, Polyakov, and Zamolodchikov predicted 1 [3,4] that many systems at criticality (at β c ) had scaling limits that were in

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“…Brownian paths (LEBPs) [17]. A version of nonintersection condition is imposed between the paths (see Eq.…”

Section: Introductionmentioning

confidence: 99%

“…(the Poisson kernels and the boundary Poisson kernels) instead of the transition probability densities [17].…”

Section: Introductionmentioning

confidence: 99%

“…Since the Brownian motions and their loop-erased parts on C are conformally invariant [17], our correlation kernel is conformally covariant. In Section V, the determinantal correlation functions in the half-infinite-strip domain R ≡ lim L→∞ R L = {z ∈ C : Re z > 0, 0 < Im z < π} given by Theorem 1 is mapped to the domain Ω = {z = re iθ ∈ C : r > 1, 0 < θ < π} (Corollary 2).…”

Section: Introductionmentioning

confidence: 99%

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Determinantal correlations of Brownian paths in the plane with nonintersection condition on their loop-erased parts

Sato

Katori

2011

Phys. Rev. E

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As an image of the many-to-one map of loop-erasing operation L of random walks, a self-avoiding walk (SAW) is obtained. The loop-erased random walk (LERW) model is the statistical ensemble of SAWs such that the weight of each SAW ζ is given by the total weight of all random walks π which are inverse images of ζ, {π : L(π) = ζ}. We regard the Brownian paths as the continuum limits of random walks and consider the statistical ensemble of loop-erased Brownian paths (LEBPs) as the continuum limits of the LERW model. Following the theory of Fomin on nonintersecting LERWs, we introduce a nonintersecting system of N -tuples of LEBPs in a domain D in the complex plane, where the total weight of nonintersecting LEBPs is given by Fomin's determinant of an N × N matrix whose entries are boundary Poisson kernels in D. We set a sequence of chambers in a planar domain and observe the first passage points at which N Brownian paths (γ 1 , . . . , γ N ) first enter each chamber, under the condition that the loop-erased parts (L(γ 1 ), . . . , L(γ N )) make a system of nonintersecting LEBPs in the domain in the sense of Fomin. We prove that the correlation functions of first passage points of the Brownian paths of the present system are generally given by determinants specified by a continuous function called the correlation kernel. The correlation kernel is of Eynard-Mehta type, which has appeared in two-matrix models and time-dependent matrix models studied in random matrix theory. Conformal covariance of correlation functions is demonstrated.

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Estimates of Random Walk Exit Probabilities and Application to Loop-Erased Random Walk

Cited by 26 publications

References 13 publications

Commutation relations for Schramm‐Loewner evolutions

Commutation relations for Schramm‐Loewner evolutions

Conformal invariance and $2D$ statistical physics

Determinantal correlations of Brownian paths in the plane with nonintersection condition on their loop-erased parts

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