arXiv subjects

Gefei Cai

Publications and source records attributed to Gefei Cai.

15 recordsLinked to original sources

arXiv2026

Monochromatic arm exponents for critical planar percolation

We determine the monochromatic three-arm exponent for critical planar percolation through the exact spectral threshold of an explicit positive integral operator family with elementary kernels, obtaining $ξ_3=0.76928375\cdots$. Unlike the monochromatic two-arm (backbone) event, the three-arm event can be obstructed by two distinct white clusters, so a Markovian exploration must retain the past connections. We introduce a two-point boundary observable that records these connections. Using the coupling between conformal loop ensemble (CLE) and Liouville quantum gravity (LQG), this observable satisfies a renewal equation in terms of quantum boundary lengths, and yields the explicit operator whose spectral information determines $ξ_3$. Our approach also gives an explicit spectral characterization of monochromatic $n$-arm exponents for all $n\ge4$.

math.PR↗

arXiv2026

The boosted loop-erased random walk

We consider a near-critical version of loop-erased random walk (LERW) by boosting the weight of the underlying random walk according to its length before loop-erasure. We construct the scaling limit in natural parametrization in the two-dimensional unit disk and the three-dimensional unit ball. The boosted scaling limit undergoes a phase transition in mutual absolute continuity with the classical LERW scaling limit; in the disk the transition occurs at boost rate $m^2=π^2/4$. As the boost increases, the curve becomes macroscopically space-filling, in the sense that unvisited cubes have side length $O(m^{-1})$, while its Hausdorff dimension at each finite boost is unchanged. These results are motivated by Makarov--Smirnov's questions on near-critical lattice models. In the planar chordal setting, we also construct a continuum loop tilt below the spectral threshold of the domain and calculate its Loewner driving function.

math.PR↗

arXiv2026

A quadratic ancestor inequality for spanning trees

We introduce a quadratic ancestor inequality for weighted spanning trees on finite undirected graphs and give two applications. First, for every finite $n$-vertex transitive graph, the expected distance in its uniform spanning tree (UST) between two independent uniform vertices is at least $\sqrt{n}-1$, resolving a conjecture in [Benjamini--Kozma, Comm. Math. Phys. 259 (2005), 257--286]. Second, we prove that the growth exponent of three-dimensional loop-erased random walk (LERW) is at least 3/2.

math.PR↗

arXiv2026

Characterization and scaling limit of three-dimensional loop-erased random walk and random-walk loop-catchers

Random-walk loop-catchers (RWLC) are a one-parameter family of random subsets of the trace of a random walk, recently introduced in [Cai 2026, arXiv:2607.18070], which interpolates between loop-erased random walk (LERW) and the full walk trace, satisfying a recovery property. We prove that the continuous version of this recovery property uniquely characterizes the scaling limit of LERW and RWLC in three dimensions, using a Green-function test via entangled multipath LERW, as outlined in [Cai 2026, arXiv:2607.18070]. For LERW, this resolves the 3D case of the conjecture of Sapozhnikov--Shiraishi [Sapozhnikov--Shiraishi 2018, Probab. Theory Related Fields 172, 615--662] and gives a new and axiomatic proof of the existence of the scaling limit, which was first proved in [Kozma 2007, Acta Math. 199, 29--152]. For RWLC, this gives the construction of 3D Brownian loop-catchers. The characterization also yields inversion invariance of the 3D LERW scaling limit.

math.PR↗

arXiv2026

The Brownian loop-catcher

We introduce a family of random connected closed subsets of planar Brownian motion, called Brownian loop-catchers, which interpolate between the continuum loop-erased random-walk (LERW) and the Brownian trace. This provides the canonical continuation of Brownian loop soup clusters to central charges $-2\le c<0$. For each such $c$, the corresponding loop-catcher satisfies the following recovery property: adding all loops from an independent Brownian loop soup of intensity $-c/2$ that intersect it recovers the Brownian trace. Furthermore, no such law exists for $c<-2$. We also show that its outer boundary is locally SLE$_κ$ with $κ= \frac{1}{3}\left(13 - c - \sqrt{(1-c)(25-c)}\right)\in[2,\frac83)$, and the probability that it intersects an interior ball of radius $\varepsilon$ is asymptotically proportional to $|\log\varepsilon|^{-1+\frac{c}{2}}$ when $-2<c<0$. Therefore, a planar Brownian trace contains an SLE$_κ$-type curve for every $κ\in[2,\frac83]$. Our construction begins with a random-walk loop-catcher on any finite graph such that recursively inserting loops from an independent random-walk loop soup to it recovers the original random walk. The key ingredient is a new entangled multipath LERW, which recovers a union of independent random-walk paths when decorated with a single common random-walk loop soup. We then prove that the random-walk loop-catcher converges to the Brownian loop-catcher under lattice approximations. To this end, we propose a novel Green function test which converts the recovery property of the Brownian loop-catcher into all mixed moments of Green functions in the remaining domain, based on the entangled multipath LERW. Consequently, the recovery property characterizes the full law of the Brownian loop-catcher, not only its filling. The Green function test also extends to the three-dimensional case.

math.PR↗

arXiv2026

Exact solution of three-point functions in critical loop models

We propose an exact formula for three-point functions on the sphere in critical loop models with primary fields $V_{(r,s)}$ characterized by $2r$ legs and a parameter $s$ that describes diagonal fields for $r=0$ and the momentum of legs for $r>0$. We demonstrate its validity in three ways: the conformal bootstrap method for 4-point functions, a transfer-matrix study of the lattice model, and a probabilistic method based on conformal loop ensemble and Liouville quantum gravity. This work provides a crucial missing piece for solving critical loop models and reveals a deep unity between three fundamental approaches to 2D statistical physics: transfer matrix, conformal field theory, and probability theory.

cond-mat.stat-mech↗

arXiv2026

Boundary four-point connectivities of conformal loop ensembles

We derive the boundary four-point Green's functions for conformal loop ensembles (CLE) with $κ\in(4,8)$. Specializing to $κ=6$ and $κ=16/3$, we establish the exact formulas for the boundary four-point connectivities in critical Bernoulli percolation and the FK-Ising model conjectured by Gori-Viti (2017, 2018). In particular, we identify a logarithmic singularity in the critical FK-Ising model. Our approach also applies to the one-bulk and two-boundary connectivities of CLE, thereby extending the factorization formula of Beliaev-Izyurov (2012) to all $κ\in(4,8)$.

math.PR↗

arXiv2026

Uniqueness of generalized conformal restriction measures and Malliavin-Kontsevich-Suhov measures for $c \in (0,1]$

In this paper, we present a unified approach to establish the uniqueness of generalized conformal restriction measures with central charge $c \in (0, 1]$ in both chordal and radial cases, by relating these measures to the Brownian loop soup. Our method also applies to the uniqueness of the Malliavin-Kontsevich-Suhov loop measures for $c \in (0,1]$, which was recently obtained in [Baverez-Jego, arXiv:2407.09080] for all $c \leq 1$ from a CFT framework of SLE loop measures. In contrast, though only valid for $c \in (0,1]$, our approach provides additional probabilistic insights, as it directly links natural quantities of MKS measures to loop-soup observables.

math.PR↗

arXiv2026

Backbone probability of planar Brownian motion

Motivated by critical planar percolation, we investigate a ``backbone'' event of planar Brownian motion, i.e.~the existence of two disjoint subpaths on the Brownian trajectory connecting the $\varepsilon$-neighborhood of the starting point to a macroscopic distance. We show that the probability of this event is $C(\log|\log\varepsilon|)^{-1}(1+o(1))$ as $\varepsilon\to0$ for some constant $C\in(0,\infty)$.

math.PR↗

arXiv2025

Three-point connectivity constant for $q$-state Potts spin clusters

Recently, Ang--Cai--Sun--Wu (2024) determined the three-point connectivity constant for two-dimensional critical percolation, confirming a prediction of Delfino and Viti (2010). In this paper, we address the analogous problem for planar critical $q$-state Potts spin clusters. We introduce a continuum three-point connectivity constant and compute it explicitly. Under the scaling-limit conjecture for Potts spin clusters, this quantity coincides with the scaling limit of the properly normalized probability that three points lie in the same spin cluster. The resulting formula agrees with the imaginary DOZZ formula up to an explicit $q$-dependent constant with a geometric interpretation. This answers a question from Delfino--Picco--Santachiara--Viti (2013). The proof exploits the coupling between CLE and LQG, together with the BCLE descriptions of $q$-state Potts scaling limits due to Miller--Sheffield--Werner (2017) and Köhler-Schindler and Lehmkühler (2025).

math.PR↗

arXiv2025

Disconnection probability of Brownian motion on an annulus

We derive an exact formula for the probability that a Brownian path on an annulus does not disconnect the two boundary components of the annulus. The leading asymptotic behavior of this probability is governed by the disconnection exponent obtained by Lawler-Schramm-Werner (2001) using the connection to Schramm-Loewner evolution (SLE). The derivation of our formula is based on this connection and the coupling with Liouville quantum gravity (LQG). As byproducts of our proof, we obtain a precise relation between Brownian motion on a disk stopped upon hitting the boundary and the SLE$_{8/3}$ loop measure on the disk; we also obtain a detailed description of the LQG surfaces cut by the outer boundary of stopped Brownian motion on a $\sqrt{8/3}$-LQG disk.

math.PR↗

arXiv2024

Integrability of Conformal Loop Ensemble: Imaginary DOZZ Formula and Beyond

The scaling limit of the probability that $n$ points are on the same cluster for 2D critical percolation is believed to be governed by a conformal field theory (CFT). Although this is not fully understood, Delfino and Viti (2010) made a remarkable prediction on the exact value of a properly normalized three-point probability. It is expressed in terms of the imaginary DOZZ formula of Schomerus, Zamolodchikov and Kostov-Petkova, which extends the structure constants of minimal model CFTs to continuous parameters. Later, similar conjectures were made for scaling limits of random cluster models and O$(n)$ loop models, representing certain three-point observables in terms of the imaginary DOZZ formula. Since the scaling limits of these models can be described by the conformal loop ensemble (CLE), such conjectures can be formulated as exact statements on CLE observables. In this paper, we prove Delfino and Viti's conjecture on percolation as well as a conjecture of Ikhlef, Jacobsen and Saleur (2015) on the nesting loop statistics of CLE. Our proof is based on the coupling between CLE and Liouville quantum gravity on the sphere, and is inspired by the fact that after reparametrization, the imaginary DOZZ formula is the reciprocal of the three-point function of Liouville CFT. Recently, Nivesvivat, Jacobsen and Ribault systematically studied a CFT with a large class of CLE observables as its correlation functions, including the ones from these two conjectures. We believe that our framework admits sufficient flexibility to exactly solve the three-point functions for CLE observables with natural geometric interpretations, including those from this CFT. As a demonstration, we solve the case corresponding to three points lying on the same loop, where the answer is a variant of the imaginary DOZZ formula.

math-ph↗

arXiv2024

SLE Loop Measure and Liouville Quantum Gravity

As recently shown by Holden and two of the authors, the conformal welding of two Liouville quantum gravity (LQG) disks produces a canonical variant of SLE curve whose law is called the SLE loop measure. In this paper, we demonstrate how LQG can be used to study the SLE loop measure. Firstly, we show that for $κ\in (8/3,8)$, the loop intensity measure of the conformal loop ensemble agrees with the SLE loop measure as defined by Zhan (2021). The former was initially considered by Kemppainen and Werner (2016) for $κ\in (8/3,4]$, and the latter was constructed for $κ\in (0,8)$. Secondly, we establish a duality for the SLE loop measure between $κ$ and $16/κ$. Thirdly, we obtain the exact formula for the moment of the electrical thickness for the shape (probability) measure of the SLE loop, which in the regime $κ\in (8/3,8)$ was conjectured by Kenyon and Wilson (2004). This relies on the exact formulae for the reflection coefficient and the one-point disk correlation function in Liouville conformal field theory. Finally, we compute several multiplicative constants associated with the SLE loop measure, which are not only of intrinsic interest but also used in our companion paper relating the conformal loop ensemble to the imaginary DOZZ formulae.

math.PR↗

arXiv2024

Quasisymmetric geometry of low-dimensional random spaces

We initiate a study of the quasisymmetric uniformization of naturally arising random fractals and show that many of them fall outside the realm of quasisymmetric uniformization to simple canonical spaces. We begin with the trace, the graph of Brownian motion, and various variants of the Schramm-Loewner evolution $\mathrm{SLE}_κ$ for $κ>0$, and show that a.s. neither is a quasiarc. After that, we study the conformal loop ensemble $\mathrm{CLE}_κ$, $κ\in (\frac{8}{3}, 4]$, and show that the collection of all points outside the loops is a.s. homeomorphic to the standard Sierpiński carpet, but not quasisymmetrically equivalent to a round carpet.

math.MG↗

arXiv2022

On Natural Measures of SLE- and CLE-Related Random Fractals

In this paper, we construct and then prove the up-to constants uniqueness of the natural measure on several random fractals, namely the SLE cut points, SLE boundary touching points, CLE pivotal points and the CLE carpet/gasket. As an application, we also show the equivalence between our natural measures defined in this paper (i.e. CLE pivotal and gasket measures) and their discrete analogs of counting measures in critical continuum planar Bernoulli percolation in [Garban-Pete-Schramm, J. Amer. Math. Soc.,2013]. Although the existence and uniqueness for the natural measure for CLE carpet/gasket have already been proved in [Miller-Schoug, arXiv:2201.01748], in this paper we provide with a different argument via the coupling of CLE and LQG.

math.PR↗