The conformal loop ensemble nesting field

Miller, Jason; Watson, Samuel S.; Wilson, David B.

doi:10.1007/s00440-014-0604-6

Cited by 17 publications

(13 citation statements)

References 17 publications

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“…Combining (8), (14) and (15), we obtain (12). There is a variance analogue of the elementary renewal theorem, see e.g.…”

Section: Define Eventmentioning

confidence: 90%

Limit theorems for critical first-passage percolation on the triangular lattice

Yao

2018

Stochastic Processes and their Applications

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Consider (independent) first-passage percolation on the sites of the triangular lattice T embedded in C. Denote the passage time of the site v in T by t(v), and assume that P (t(v) = 0) = P (t(v) = 1) = 1/2. Denote by b0,n the passage time from 0 to the halfplane {v ∈ T : Re(v) ≥ n}, and by T (0, nu) the passage time from 0 to the nearest site to nu, where |u| = 1. We prove that as n → ∞, b0,n/ log n → 1/(2 √ 3π) a.s., E[b0,n]/ log n → 1/(2 √ 3π) and Var[b0,n]/ log n → 2/(3This answers a question of Kesten and Zhang (1997) and improves our previous work (2014). From this result, we derive an explicit form of the central limit theorem for b0,n and T (0, nu). A key ingredient for the proof is the moment generating function of the conformal radii for conformal loop ensemble CLE6, given by Schramm, Sheffield and Wilson (2009).

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“…Combining (8), (14) and (15), we obtain (12). There is a variance analogue of the elementary renewal theorem, see e.g.…”

Section: Define Eventmentioning

confidence: 90%

Limit theorems for critical first-passage percolation on the triangular lattice

Yao

2018

Stochastic Processes and their Applications

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“…Also sample an independent uniformly distributed sign σ Γ P t´1, 1u. Then, for a well-chosen constant λ ą 0 (equal to a π{8 in our normalisation of the GFF), the field ÿ ΓPCLE4 1 IntpΓq pφ Γ`σΓ 2λq is a Gaussian free field φ D in D with zero boundary conditions (see also [36,26]). In this coupling the CLE 4 loops can be interpreted as level lines of the continuum Gaussian free field φ D on D and are in fact deterministic functions of the GFF ( [25], see also [2]).…”

Section: By Takingmentioning

confidence: 99%

The random pseudo-metric on a graph defined via the zero-set of the Gaussian free field on its metric graph

Lupu¹,

Werner²

2017

Probab. Theory Relat. Fields

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Abstract. We further investigate properties of the Gaussian free field (GFF) on the metric graph associated to a discrete weighted graph (where the edges of the latter are replaced by continuous line-segments of appropriate length) that has been introduced by the first author. On such a metric graph, the GFF is a random continuous function that generalises one-dimensional Brownian bridges so that one-dimensional techniques can be used.In the present paper, we define and study the pseudo-metric defined on the metric graph (and therefore also on the discrete graph itself), where the length of a path on the metric graph is defined to be the local time at level zero accumulated by the Gaussian free field along this path.We first derive a pathwise transformation that relates the GFF on the metric graph with the reflected GFF on the metric graph via the pseudo-distance defined by the latter. This is a generalisation of Paul Lévy's result relating the local time at zero of Brownian motion to the supremum of another Brownian motion. We also compute explicitly the distribution of certain functionals of this pseudo-metric and of the GFF. In particular, we point out that when the boundary consists of just two points, the law of the pseudo-distance between them depends solely on the resistance of the network between them.We then discuss questions related to the scaling limit of this pseudo-metric in the two-dimensional case, which should be the conformally invariant way to measure distances between CLE(4) loops introduced and studied by the second author with Wu, and by Sheffield, Watson and Wu. Our explicit laws on metric graphs also lead to new conjectures for related functionals of the continuum GFF on fairly general Riemann surfaces.

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“…In fact, it is straightforward to see that C and α (which in principle depend on κ) may be chosen uniformly for κ ∈ [4, 6] (say). Indeed, it follows from the proof in [MWW15] that they depend only on the law of the log conformal radius of the outermost loop containing 0 for a CLE κ in D, and this varies continuously in κ, [SSW09]. Hence, the result follows by letting R → ∞ in Lemma 2.27 and noting that the second smallest loop containing D is contained in rD with arbitarily high probability as r → ∞, uniformly in κ.…”

Section: Whole Plane Cle and Conclusion Of The Proofsmentioning

confidence: 90%

“…For fixed κ ∈ [4, 8), let Γ C , Γ RD denote whole plane CLE κ and CLE κ on RD respectively. The key to this lemma is Theorem 9.1 in [MWW15], which states (in particular) that Γ RD rapidly converges to Γ C in the following sense. For some C, α > 0, Γ RD and Γ C can be coupled so that for any r > 0 and R > r, with probability at least 1 − C(R/r) −α , there is a conformal map ϕ from some D ⊃ (R/r) 1/4 D to D ⊃ (R/r) 1/4 D, which maps the nested loops of Γ RD -starting with the smallest containing rD -to the corresponding nested loops of Γ C , and has low distortion in the sense that |ϕ (z)…”

Section: Whole Plane Cle and Conclusion Of The Proofsmentioning

confidence: 99%

Mating of trees for critical Liouville quantum gravity

Aru¹,

Holden²,

Powell³

et al. 2021

Preprint

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In a groundbreaking work, Duplantier, Miller and Sheffield showed that subcritical Liouville quantum gravity (LQG) coupled with Schramm-Loewner evolutions (SLE) can be described by the mating of two continuum random trees. In this paper, we consider the counterpart of their result for critical LQG and SLE, i.e., for the case when γ 2 = κ = 16/κ = 4. We prove that as one sends κ ↓ 4 in the subcritical setting, the space-filling SLE κ in a disk degenerates to the CLE 4 exploration introduced by Werner and Wu, along with a collection of i.i.d. coin tosses indexed by the branch points of the exploration. Furthermore, in the κ = 16/γ 2 ↓ 4 limit, the pair of continuum random trees collapse into a single continuum random tree, and we observe that upon applying an appropriate affine transform to the encoding Brownian motions before taking the limit, we get convergence to a pair of independent Brownian motions (A, B). The Brownian motion A encodes the LQG distance from the CLE loops to the boundary of the disk, while the Brownian motion B encodes the boundary lengths of the CLE 4 loops. In contrast to the subcritical setting, (A, B) does not determine the CLE-decorated LQG surface. Contents4 Liouville quantum gravity and mating of trees 4.1 Liouville quantum gravity .

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The conformal loop ensemble nesting field

Cited by 17 publications

References 17 publications

Limit theorems for critical first-passage percolation on the triangular lattice

Limit theorems for critical first-passage percolation on the triangular lattice

The random pseudo-metric on a graph defined via the zero-set of the Gaussian free field on its metric graph

Mating of trees for critical Liouville quantum gravity

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