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Extremal Distance and Conformal Radius of a CLE 4 Loop Juhan Aru, Titus Lupu, Avelio Sepúlveda

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Extremal Distance and Conformal Radius of a CLE 4 Loop Juhan Aru, Titus Lupu, Avelio Sepúlveda Extremal distance and conformal radius of a CLE_4 loop Juhan Aru, Titus Lupu, Avelio Sepúlveda To cite this version: Juhan Aru, Titus Lupu, Avelio Sepúlveda. Extremal distance and conformal radius of a CLE_4 loop. 2020. hal-02557456v2 HAL Id: hal-02557456 https://hal.archives-ouvertes.fr/hal-02557456v2 Preprint submitted on 10 Jun 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. EXTREMAL DISTANCE AND CONFORMAL RADIUS OF A CLE4 LOOP JUHAN ARU, TITUS LUPU, AND AVELIO SEPÚLVEDA Abstract. Consider CLE4 in the unit disk and let ` be the loop of the CLE4 surrounding the origin. Schramm, Sheffield and Wilson determined the law of the conformal radius seen from the origin of the domain surrounded by `. We complement their result by determining the law of the extremal distance between ` and the boundary of the unit disk. More surprisingly, we also compute the joint law of these conformal radius and extremal distance. This law involves first and last hitting times of a one-dimensional Brownian motion. Similar techniques also allow us to determine joint laws of some extremal distances in a critical Brownian loop-soup cluster. 1. Introduction The Conformal Loop Ensembles CLEκ, κ 2 (8=3; 4] form a one-parameter family of random collections of simple loops in a simply connected planar domain that are conformally invariant in law. They were introduced in [She09, SW12] as the conjectural scaling limits of the interfaces for various statistical physics models. This conjecture has been confirmed only for some values of the parameter κ. In particular for κ = 4, it is known that the CLE4 appears in the scaling limit of the outer boundaries of outermost sign components of a metric graph GFF [Lup19] and thereby has a natural coupling to the 2D continuum Gaussian free field (GFF) [MS11, WW16, ASW17]. Furthermore, the CLE4 is conjectured to be scaling limit of loops in the double-dimerp model [Ken14, Dub19, BC18] and in the loop O(n) model with parameters n = 2; x = 1= 2 [KN04, PS17]. The main result of this paper concerns the geometry of the loop of a CLE4 surrounding the origin. We compute the joint law of the extremal distance from this loop to the boundary of the domain, together with the conformal radius seen from the origin of the domain surrounded by the loop. The conformal radii of CLEκ for κ 2 (8=3; 4] have been already identified in [She09], by a different but related method, to first hitting times of particular diffusions. In [SSW09] the explicit density of their laws has been further computed. No identification of the law of the extremal distance has appeared so far. Our approach for κ = 4 uses the coupling with the GFF. The laws that appear are related to certain random times of the Brownian motion. For B a standard Brownian motion, define (1.1) T := infft ≥ 0 : jBtj = πg; τ := supf0 ≤ t ≤ T : Bt = 0g: Given two closed sets C1, C2, denote the extremal distance between them by ED(C1;C2). See Section 2.1 for details. The main theorem of this paper is as follows (see also Figure 1). Theorem 1.1. Let ` be the loop of a CLE4 in the unit disk D surrounding the origin. Let CR(0; Dn`) be the conformal radius of the origin in the domain surrounded by `. Then, the law of (2π ED(`; @D); − log CR(0; Dn`)) equals that of (τ; T ). Date: 2010 Mathematics Subject Classification. 31A15; 60G15; 60G60; 60J65; 60J67; 81T40. Key words and phrases. Brownian motion; conformal loop ensemble; Gaussian free field; isomorphism theorems; local set; loop-soup; Schramm-Loewner evolution. 1 ` π T 0 τ −π Figure 1. Graphic explanation of Theorem 1.1. The dashed line represents 2π ED(`; @D) and the continuous lines represents − log CR(0; Dn`). Note that in particular a.s. 2π ED(`; @D) ≤ − log CR(0; Dn`). This is actually a deterministic inequality, satisfied by any simple loop disconnecting 0 from @D (see Corollary 2.4), and the equality holds if and only if the loop is a circle centered at 0. Using standard distortion bounds one can deduce from Theorem 1.1 some information on the size and shape the CLE4 loop ` surrounding 0, and in particular how far it is from being a circle centered at 0. Corollary 1.2. Let ` be the loop of a CLE4 in the unit disk D surrounding the origin. Denote r−(`) := d(0; `); r+(`) := maxfjzj : z 2 `g: Then one has the following exponents: log (r (`)−1 > R) log (r (`)−1 > R) 1 lim P + = lim P − = − ; R!+1 log R R!+1 log R 8 and log (r (`)=r (`) > R) 1 (1.2) lim P + − = − : R!+1 log R 2 Moreover, (1.2) also holds if ` is distributed according to the stationary CLE4 distribution in C introduced in [KW16]. The exponent for r−(`) has been known since [SSW09] and is related to the dimension of the CLE4 gasket. The exponents for r+(`) and r+(`)=r−(`) have not appeared in other works, however it is possible that they may also be computable by different methods. A generalization of Theorem 1.1 describes the geometry of a cluster of a Brownian loop-soup [LW04] involved in the construction of the CLE4 as in [SW12]. To state the theorem, let us introduce two further random times related to the Brownian motion B T := infft ≥ T : Bt = 0g; τ := supf0 ≤ t ≤ T − T : Bt+T = BT g: We can now state our second main theorem (see also Figure 2). Theorem 1.3. Take a critical Brownian loop-soup in D and let C be the outer-most loop-soup cluster surrounding the origin. Let `o and `i denote the outer and inner boundary of C w.r.t. the origin, respectively. Then, o i o i i (1) The quadruple (2π ED(@D; ` ); − log CR(0; Dn` ); 2π ED(` ; ` ); − log CR(0; Dn` ))) is equal in law to (τ; T; τ; T ). o o i i i (2) Furthermore, the quadruple (2π ED(@D; ` ); 2π ED(` ; ` ); 2π ED(@D; ` ); − log CR(0;Dn` )) has the same law as (τ; τ; τ + T; T ). 2 o ` π T τ¯ `i T + τ¯ 0 τ T¯ −π Figure 2. Graphic explanation of Theorem 1.3. The dashed orange, resp. cyan, o i line represents 2π ED(` ;@D), 2π ED(` ;@D), the continuous orange, resp. cyan, line o i represents − log CR(0; Dn` ), − log CR(0; Dn` ) and the doted magenta line repre- o i o sents 2π ED(` ; ` ). Note that in this case, the joint law of − log(CR(0; Dn` )) and i 2π ED(` ;@D)) is not computed in Theorem 1.3 . Notice that in both cases, we have only given the joint law for a quadruple of (reduced) extremal distances. The reason is the following: o i • The joint law of (− log CR(0; Dn` ); 2π ED(@D; ` ))) cannot be naturally obtained from the same Brownian motion B, i.e. it is not the same as the law of (T; τ + T ). o i Indeed, if the joint law were equal to that of (T; τ + T ), then − log CR(0; Dn` ) < 2π ED(@D; ` ) would hold almost surely. However, this cannot be true as the following happens with positive 0 o probability: the conformal radius seen from ` is very small, hence − log CR(0; Dn` ) very large, i and at the same time the extremal distance ED(@D; ` ) is very small. See Figure 3 for an explanation. Let us now give an idea of the proof of this result. The first step is to couple the CLE4 (Miller- Sheffield coupling) and the critical Brownian loop-soup to the continuum GFF. We will use the notion of local sets of the GFF [SS13, WP20] together with some recent developments in this topic [ASW17, ALS20a, ALS20b]. In this language CLE4 is described by a two-valued local set, denoted A−2λ,2λ. The connection between the 2D continuum GFF and the 1D Brownian motion is obtained from the exploration of the GFF by local sets. Indeed, let (ηt)t≥0 be a continuous growing family of local sets of a GFF in D and hηt (0) be the harmonic extension in 0 of the values of the GFF discovered `i `o Figure 3. Main reason why we cannot compute the joint law of o i i (− log CR(0; Dn` ); 2π ED(@D; ` ))): it is possible that ` is arbitrarily close to the boundary, while `o is arbitrarily close to the center. 3 by ηt. Then the process (hηt (0))t≥0 is a continuous local martingale, and thus a time changed Brownian motion. Moreover, the natural time parametrization is such that at each time t, we have −1 −(2π) log CR(z; Dnηt) = t. In [ASW17], the authors provide a construction of the local set A−2λ,2λ (that has the law of CLE4) by a growing family of local sets. This allows them to give a new proof of the fact that − log(CR(0; Dn`)) has the law of the exit time of a Brownian motion [ASW17] from the interval [−π; π], reproving the results of [She09, SSW09] for κ = 4 in a slightly different, but connected way: namely the Brownian motion entering the coupling with the GFF is very much related to the Brownian motion for the case κ = 4 in [She09, SSW09].
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