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The Conformal Loop Ensemble Nesting Field The conformal loop ensemble nesting field The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Miller, Jason and Samuel S. Watson, and David B. Wilson."The conformal loop ensemble nesting field." Probability Theory and Related Fields, vol. 163, no. 3, March 2015, pp. 769-801. As Published http://dx.doi.org/10.1007/s00440-014-0604-6 Publisher Springer-Verlag Version Author's final manuscript Citable link http://hdl.handle.net/1721.1/104898 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Probab. Theory Relat. Fields (2015) 163:769–801 DOI 10.1007/s00440-014-0604-6 The conformal loop ensemble nesting field Jason Miller · Samuel S. Watson · David B. Wilson Received: 16 January 2014 / Revised: 17 November 2014 / Published online: 25 March 2015 © Springer-Verlag Berlin Heidelberg 2015 Abstract The conformal loop ensemble CLEκ with parameter 8/3 <κ<8isthe canonical conformally invariant measure on countably infinite collections of non- crossing loops in a simply connected domain. We show that the number of loops surrounding an ε-ball (a random function of z and ε) minus its expectation converges almost surely as ε → 0 to a random conformally invariant limit in the space of distributions, which we call the nesting field. We generalize this result by assigning i.i.d. weights to the loops, and we treat an alternate notion of convergence to the nesting field in the case where the weight distribution has mean zero. We also establish estimates for moments of the number of CLE loops surrounding two given points. Keywords SLE · CLE · Conformal loop ensemble · Gaussian free field Mathmatics Subject Classifications Primary 60J67 · 60F10; Secondary 60D05 · 37A25 Contents 1 Introduction ............................................. 770 2 Basic CLE estimates ........................................ 774 3 Co-nesting estimates ........................................ 777 4 Regularity of the ε-ball nesting field ................................ 779 5 Properties of Sobolev spaces .................................... 789 J. Miller Massachusetts Institute of Technology and Microsoft Research, Cambridge, MA, USA S. S. Watson Massachusetts Institute of Technology, Cambridge, MA, USA D. B. Wilson (B) Microsoft Research, Redmond, WA, USA e-mail: [email protected] 123 770 J. Miller et al. 6 Convergence to limiting field .................................... 792 7 Step nesting ............................................. 795 8 Further questions .......................................... 797 9 Appendix A: Rapid convergence to full-plane CLE ........................ 798 10 Notation ............................................... 800 References ................................................ 801 1 Introduction The conformal loop ensemble CLEκ for κ ∈ (8/3, 8) is the canonical conformally invariant measure on countably infinite collections of non-crossing loops in a simply connected domain D C[12,14]. It is the loop analogue of SLEκ , the canonical conformally invariant measure on non-crossing paths. Just as SLEκ arises as the scaling limit of a single interface in many two-dimensional discrete models, CLEκ is a limiting law for the joint distribution of all of the interfaces. Figures 1 and 2 shows two discrete loop models believed or known to have CLEκ as a scaling limit. Figure 3 illustrates these scaling limits for several values of κ. Let κ ∈ (8/3, 8),letD Cbe a simply connected domain, and let be a CLEκ in D. For each point z ∈ D and ε>0, we let Nz(ε) be the number of loops of which surround B(z,ε), the ball of radius ε centered at z. We prove the existence and conformal invariance of the limit as ε → 0 of the random function z → Nz(ε) − E[Nz(ε)] (with no additional normalization) in an appropriate space of distributions (Theorem 1.1). We refer to this object as the nesting field because, roughly, its value describes the fluctuations of the nesting of around its mean. This result also holds when the loops are assigned i.i.d. weights. More precisely, we fix a probability measure μ on R with finite second moment, define z(ε) to be the set of loops in surrounding B(z,ε), and define Sz(ε) = ξL, (1.1) L∈z (ε) (a) (b) (c) Fig. 1 Nesting of loops in the O(n) loop model. Each O(n) loop configuration has probability proportional to xtotal length of loops × n# loops. For a certain critical value of x,theO(n) model for 0 ≤ n ≤ 2hasa “dilute phase”, which is believed to converge CLEκ for 8/3 <κ≤ 4 with n =−2cos(4π/κ).Forx above this critical value, the O(n) loop model is in a “dense phase”, which is believed to converge to CLEκ for 4 ≤ κ ≤ 8, again with n =−2cos(4π/κ).See[6] for further background, a site percolation, b O(n) loop model. Percolation corresponds to n = 1andx = 1, which is in the dense phase, c area shaded by nesting of loops 123 The CLE nesting field 771 (a) (b) (c) Fig. 2 Nesting of loops separating critical Fortuin-Kasteleyn (FK) clusters from dual clusters. Each FK ( /( − ))# edges × #clusters bond configuration has probability proportional√ to p 1 p q [4], where there is = /( + / ) ≥ ≤ ≤ believed to be a critical point at p 1 1 1 q (proved for q 1[2]). For 0 √q 4, these loops are believed to have the same large-scale behavior as the O(n) model loops for n = q in the dense phase, that is, to converge to CLEκ for 4 ≤ κ ≤ 8(see[6,11]), a Critical FK bond configuration. Here q = 2, b loops separating FK clusters from dual clusters, c area shaded by nesting of loops where ξL are i.i.d. random variables with law μ. We show that z → Sz(ε) − E[Sz(ε)] converges as ε → 0 to a distribution we call the weighted nesting field. When κ = 4 and μ is a signed Bernoulli distribution, the weighted nesting field is the GFF [9,10]. Our result serves to generalize this construction to other values of κ ∈ (8/3, 8) and weight measures μ. In Theorem 1.2, we answer a question asked in [12, Problem 8.2]. The weighted nesting field is a random distribution, or generalized function, on D. Informally, it is too rough to be defined pointwise on D, but it is still possible to integrate it against sufficiently smooth compactly supported test functions on D.More precisely, we prove convergence to the nesting field in a certain local Sobolev space s ( ) ⊂ ∞( ) ∞( ) Hloc D Cc D on D, where Cc D is the space of compactly supported smooth , ∞( ) ∈ R functions on D Cc D is the space of distributions on D, and the index s is a parameter characterizing how smooth the test functions need to be. We review all the relevant definitions in Sect. 5. The nesting field gives a loop-free description of the conformal loop ensemble. For κ ≤ 4 we believe that the nesting field determines the CLE, but that for κ>4the CLE contains more information. (See Question 2 in the open problems section.) In order to prove the existence of the nesting field, we show that the law of CLE near a point rapidly forgets loops that are far away, in a sense that we make quantitative. ∈ ∞( ) ∈ ∞( ) , Given h Cc D and f Cc D , we denote by h f the evaluation of the ◦ ϕ−1 ∈ ∞( ) linear functional h at f . Recall that the pullback h of h Cc D under a ϕ−1 ◦ ϕ−1, := , |ϕ |2 ◦ ϕ ∈ ∞(ϕ( )) conformal map is defined by h f h f for f Cc D . Theorem 1.1 Fix κ ∈ (8/3, 8) and δ>0, and suppose μ is a probability measure on R with finite second moment. Let D Cbe a simply connected domain. Let be a CLEκ on D and (ξL)L∈ be i.i.d. weights on the loops of drawn from the distribution μ. Recall that for ε>0 and z ∈ D, Sz(ε) denotes Sz(ε) = ξL. L∈ L surrounds B(z,ε) 123 772 J. Miller et al. (a) (b) (c) (d) Fig. 3 Simulations of discrete loop models which converge to (or are believed to converge to, indicated with asterisk)CLEκ in the fine mesh limit. For each of the CLEκ ’s, one particular nested sequence of loops is outlined. For CLEκ , almost all of the points in the domain are surrounded by an infinite nested sequence of loops, though the discrete samples shown here display only a few orders of nesting, a CLE3 (from critical Ising model), b CLE4 (from the FK model with q = 4) , c CLE16/3 (from the FK model with q = 2), d CLE6 (from critical bond percolation) Let hε(z) = Sz(ε) − E[Sz(ε)]. (1.2) −2−δ( ) = (, (ξ )) There exists an Hloc D -valued random variable h h L such that for ∈ ∞( ) , = , (, (ξ )) all f Cc D , almost surely limε→0 hε f h f . Moreover, h L is almost surely a deterministic conformally invariant function of the CLE and the loop weights (ξL)L∈: almost surely, for any conformal map ϕ from D to another simply connected domain, we have −1 h(ϕ(), (ξϕ(L))L∈) = h(, (ξL)L∈) ◦ ϕ . 123 The CLE nesting field 773 In Theorem 6.2, we prove a stronger form of convergence, namely almost sure convergence in the norm topology of H −2−δ(D), when ε tends to 0 along any given geometric sequence. We also consider the step nesting sequence, defined by n n h ( ) = ξ − E ξ , ∈ N, n z Lk (z) Lk (z) n k=1 k=1 where the random variables (ξL)L∈ are i.i.d.
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