Dapeng Zhan
We prove that, for $κ\in(0,4)$ and $ρ\ge (κ-4)/2$, the chordal SLE$(κ;ρ)$ trace started from $(0;0^+)$ or $(0;0^-)$ satisfies the reversibility property. And we obtain the equation for the reversal of the chordal SLE$(κ;ρ)$ trace started from $(0;b_0)$, where $b_0>0$.
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We prove that the chordal SLE$_κ$ trace is reversible for $κ\in(0,4]$.
We show that, for $κ\le 4$, the integral of the two-sided radial SLE$_κ$ measures over all interior points is chordal SLE$_κ$ biased by the path's natural length, which is its $(1+κ/8)$-dimensional Minkowski content.
We prove that the chordal SLE$_κ$ trace is reversible for $κ\in(0,4]$.
We derive some geometric properties of chordal SLE$(κ;\vecρ)$ processes. Using these results and the method of coupling two SLE processes, we prove that the outer boundary of the final hull of a chordal SLE$(κ;\vecρ)$ process has the same distribution as the image of a chordal SLE$(κ';\vec{ρ'})$ trace, where $κ>4$, $κ'=16/κ$, and the forces $\vecρ$ and $\vec{ρ'}$ are suitably chosen. We find that for $κ\ge 8$, the boundary of a standard chordal SLE$(κ)$ hull stopped on swallowing a fixed $x\in\R\sem\{0\}$ is the image of some SLE$(16/κ;\vecρ)$ trace started from $x$. Then we obtain a new proof…
We improve the geometric properties of SLE$(κ;\vecρ)$ processes derived in an earlier paper, which are then used to obtain more results about the duality of SLE. We find that for $κ\in (4,8)$, the boundary of a standard chordal SLE$(κ)$ hull stopped on swallowing a fixed $x\in\R\sem\{0\}$ is the image of some SLE$(16/κ;\vecρ)$ trace started from a random point. Using this fact together with a similar proposition in the case that $κ\ge 8$, we obtain a description of the boundary of a standard chordal SLE$(κ)$ hull for $κ>4$, at a finite stopping time. Finally, we prove that for $κ>4$, in many c…
In the paper we prove that, for $κ\in(0,8)$, the $n$-point boundary Green's function of exponent $\frac8κ-1$ for chordal SLE$_κ$ exists. We also prove that the convergence is uniform over compact sets and the Green's function is continuous. We also give up-to-constant bounds for the Green's function.
We give a new proof of the reversibility of the Schramm Loewner evolution for $κ\leq 4$. The main ideas used in the proof are similar to those used in the original proof of this result, given by Zhan.
We derive a number of estimates for the probability that a chordal SLE path in the upper half plane H intersects a semicircle centred on the real line. We prove that if 0<kappa<8 and gamma:[0,infinity) to closure(H) is a chordal SLE in H from 0 to infinity, then P(gamma[0,infinity) cap C(x;rx) neq emptyset) asymp r^(4a-1) where a=2/kappa and C(x;rx) denotes the semicircle centred at x>0 of radius rx, 0<r<1/3, in the upper half plane. As an application of our results, for 0<kappa<8, we derive an estimate for the diameter of a chordal SLE path in H between two real boundary points 0 and x>0. For…
Decomposition of Hypergeometric SLE and Reversibility
本文研究了κ∈(4,8)的超几何SLE过程，证明其条件律可由SLE_κ(ρ_)描述，并据此构造曲线对推导出其可逆性。
In this paper we consider the natural random walk on a planar graph and scale it by a small positive number $δ$. Given a simply connected domain $D$ and its two boundary points $a$ and $b$, we start the scaled walk at a vertex of the graph nearby $a$ and condition it on its exiting $D$ through a vertex nearby $b$, and prove that the loop erasure of the conditioned walk converges, as $δ\downarrow 0$, to the chordal SLE$_{2}$ that connects $a$ and $b$ in $D$, provided that an invariance principle is valid for both the random walk and the dual walk of it.
For a chordal SLE$_κ$ ($κ\in(0,8)$) curve in a domain $D$, the $n$-point Green's function valued at distinct points $z_1,\dots,z_n\in D$ is defined to be $$G(z_1,\dots,z_n)=\lim_{r_1,\dots,r_n\downarrow 0} \prod_{k=1}^n r_k^{d-2} \mathbb{P}[\mbox{dist}(γ,z_k)<r_k,1\le k\le n],$$ where $d=1+\fracκ{8}$ is the Hausdorff dimension of SLE$_κ$, provided that the limit converges. In this paper, we will show that such Green's functions exist for any finite number of points. Along the way we provide the rate of convergence and modulus of continuity for Green's functions as well. Finally, we give up-to-…
The main result of this paper is that, for $κ\in(0,4]$, whole-plane SLE$_κ$ satisfies reversibility, which means that the time-reversal of a whole-plane SLE$_κ$ trace is still a whole-plane SLE$_κ$ trace. In addition, we find that the time-reversal of a radial SLE$_κ$ trace for $κ\in(0,4]$ is a disc SLE$_κ$ trace with a marked boundary point. The main tool used in this paper is a stochastic coupling technique, which is used to couple two whole-plane SLE$_κ$ traces so that they overlap. Another tool used is the Feynman-Kac formula, which is used to solve a PDE. The solution of this PDE is then …
This paper concerns a random walk on a planar graph and presents certain estimates concerning the harmonic measures for the walk in a grid domain which estimates are useful for showing the convergence of a LERW (loop-erased random walk) to an SLE (stochastic Loewner evolution). We assume that the walk started at a fixed vertex of the graph satisfies the invariance principle as in Yadin and Yehudayoff [16] in which the convergence of LERW to a radial SLE is established in this setting. Our main concern is chordal case, where a random walk is started at a boundary vertex of a simply connected gr…
We prove that, for $κ\le 4$, backward chordal SLE$_κ$ admits backward chordal SLE$_κ(-4,-4)$ decomposition for the capacity parametrization. This means that, for any bounded measurable subset $U\subset Q_4:={\mathbb R}_+\times{\mathbb R}_-$, if we integrate the laws of extended backward chordal SLE$_κ(-4,-4)$ with different pairs of force points $(x,y)$ against some suitable density function $G(x,y)$ restricted to $U$, then we get a measure, which is absolutely continuous with respect to the law of backward chordal SLE$_κ$, and the Radon-Nikodym derivative is a constant depending on $κ$ times …
Let $κ\in(4,8)$. Let $γ$ be an SLE$_κ$ curve in a Jordan domain $D$ connecting $a_1\ne a_2\in\partial D$. We attach $γ$ with two open boundary arcs $A_1,A_2$ of $D$, which share end points $b_1\ne b_2\in\partial D\setminus\{a_1,a_2\}$, and consider for each $z_0\in D$ the limit $$ \lim_{r \downarrow 0}r^{1-\frac 38κ} \mathbb{P}[γ\cup A_1\cup A_2 \mbox{ has a cut point in }\{|z-z_0|<r\}].$$ We prove that the limit converges, derive a rate of convergence, and obtain the exact formula of the limit up to a multiplicative constant depending only on $κ$.
A deterministic proof of Loewner energy reversibility via local reversals
该研究提出弦Loewner能量方向可逆性的纯确定性新证明，通过局部分增量反转弦方向实现，关键证明了过固定有限点集的最小能量弦是分段双曲测地线，思路借鉴SLE可逆性证明但无需概率方法。
For $κ\in(0,4]$, a family of annulus SLE$(κ;Λ)$ processes were introduced in [14] to prove the reversibility of whole-plane SLE$(κ)$. In this paper we prove that those annulus SLE$(κ;Λ)$ processes satisfy a restriction property, which is similar to that for chordal SLE$(κ)$. Using this property, we construct $n\ge 2$ curves crossing an annulus such that, when any $n-1$ curves are given, the last curve is a chordal SLE$(κ)$ trace.
We show that, for $κ\in(0,8)$, the integral of the laws of two-sided radial SLE$_κ$ curves through different interior points against a measure with SLE$_κ$ Green function density is the law of a chordal SLE$_κ$ curve, biased by the path's natural length. We also show that, for $κ>0$, the integral of the laws of extended SLE$_κ(-8)$ curves through different interior points against a measure with a closed formula density restricted in a bounded set is the law of a chordal SLE$_κ$ curve, biased by the path's capacity length restricted in that set. Another result is that, for $κ\in(4,8)$, if one i…
We give a direct construction of the conformally invariant measure on self-avoiding loops in Riemann surfaces (Werner measure) from chordal $\text{SLE}_{8/3}$. We give a new proof of uniqueness of the measure and use Schramm's formula to construct a measure on boundary bubbles encircling an interior point. After establishing covariance properties for this bubble measure, we apply these properties to obtain a measure on loops by integrating measures on boundary bubbles. We calculate the distribution of the conformal radius of boundary bubbles encircling an interior point and deduce from it expl…
Onsager--Machlup functional for multiple SLEs
本文将多弦 Loewner 势解释为多弦 SLE 的 Onsager--Machlup 泛函，并将共形形变公式推广至多径向 Loewner 势，导出了多径向 SLE 的 Onsager--Machlup 泛函。
We investigate the relation between the Laplacian-$b$ motion and stochastic Komatu-Loewner evolution (SKLE) on multiply connected subdomains of the upper half-plane, both of which are analogues to SLE. In particular, we show that, if the driving function of an SKLE is given by a certain stochastic differential equation, then this SKLE is the same as a time-changed Laplacian-$b$ motion. As an application, we prove the finite time explosion of SKLE corresponding to Laplacian-$0$ motion, or $\mathrm{SLE_6}$, in the sense that the solution to the Komatu-Loewner equation for the slits blows up.
Time-reversal of multiple-force-point chordal $\mathrm{SLE}_κ(\underlineρ)$
本文运用Miller和Sheffield的虚几何技术证明了Zhan提出的一般情形下多力点弦向SLE_κ(ρ)时间反转分布猜想，明确了其时间反转分布的绝对连续性及Radon-Nikodym导数形式。
We first prove that, for $κ\in(0,4)$, a whole-plane SLE$(κ;κ+2)$ trace stopped at a fixed capacity time satisfies reversibility. We then use this reversibility result to prove that, for $κ\in(0,4)$, a chordal SLE$_κ$ curve stopped at a fixed capacity time can be mapped conformally to the initial segment of a whole-plane SLE$(κ;κ+2)$ trace. A similar but weaker result holds for radial SLE$_κ$. These results are then used to study the ergodic behavior of an SLE curve near its tip point at a fixed capacity time. The proofs rely on the symmetry of backward SLE laminations and conformal removabilit…
We study some features of the energy of a deterministic chordal Loewner chain, which is defined as the Dirichlet energy of its driving function. In particular, using an interpretation of this energy as a large deviation rate function for SLE$_κ$ as $κ$ tends to 0 and the known reversibility of the SLE$_κ$ curves for small $κ$, we show that the energy of a deterministic curve from one boundary point A of a simply connected domain D to another boundary point B, is equal to the energy of its time-reversal ie. of the same curve but viewed as going from B to A in D.
We consider a particular type of $\sqrt{8/3}$-Liouville quantum gravity surface called a doubly marked quantum disk (equivalently, a Brownian disk) decorated by an independent chordal SLE$_6$ curve $η$ between its marked boundary points. We obtain descriptions of the law of the quantum surfaces parameterized by the complementary connected components of $η([0,t])$ for each time $t \geq 0$ as well as the law of the left/right $\sqrt{8/3}$-quantum boundary length process for $η$.
We aim at finding the reversal of radial SLE and proving the reversibility of whole-plane SLE. For this purpose, we define annulus SLE$(κ,Λ)$ processes in doubly connected domains with one marked boundary point. We derive some partial differential equation for $Λ$, which is sufficient for the annulus SLE$(κ,Λ)$ process to satisfy commutation relation. If $Λ$ satisfies this PDE, then using a coupling technique, we are able to construct a global commutation coupling of two annulus SLE$(κ,Λ)$ processes. If more conditions are satisfied, the coupling exists in the degenerate case, which becomes a …
The Green's function for the chordal Schramm-Loewner evolution $SLE_κ$ for $0 < κ< 8$, gives the normalized probability of getting near points. We give up-to-constant bounds for the two-point Green's function.
Reversibility of whole-plane SLE for $κ> 8$
本文证明了κ>8时全平面SLE$_κ$的可逆性，解决了Viklund和Wang猜想的最后情形，其证明依赖于一个新的树配对定理，该定理涉及由独立径向空间填充SLE曲线装饰的圆盘上的Liouville量子引力。
We give estimates for the probability that a chordal, radial or two-sided radial SLE$_κ$ curve retreats far from its terminal point after coming close to it, for $κ\leq 4$. The estimates are uniform over all initial segments of the curve, and are sharp up to a universal constant.
We review some recently completed research that establishes the scaling limit of Fomin's identity for loop-erased random walk on Z^2 in terms of the chordal Schramm-Loewner evolution (SLE) with parameter 2. In the case of two paths, we provide a simplified proof of the identity for loop-erased random walk and simple random walk, and prove directly that the corresponding identity holds for chordal SLE(2) and Brownian motion. We also include a brief introduction to SLE and discussion of the relationship between SLE(2) and loop-erased random walk.
The purpose of this note is to describe a framework which unifies radial, chordal and dipolar SLE. When the definition of SLE(kappa;rho) is extended to the setting where the force points can be in the interior of the domain, radial SLE(kappa) becomes chordal SLE(kappa;rho), with rho=kappa-6, and vice versa. We also write down the martingales describing the Radon-Nykodim derivative of SLE(kappa;rho_1,...,rho_n) with respect to SLE(kappa).
Multiple chordal SLE(0) and classical Calogero-Moser system
本文发展了(n,m)型多重弦SLE(0)系统的一般理论，将其扩展至n=2m情形之外，并证明其在特定单值化下的迹对应于实有理函数的实轨迹，且Loewner动力学遵循经典Calogero-Moser哈密顿量演化。
An annulus SLE$_κ$ trace tends to a single point on the target circle, and the density function of the end point satisfies some differential equation. Some martingales or local martingales are found for annulus SLE$_4$, SLE$_8$ and SLE$_{8/3}$. From the local martingale for annulus SLE$_4$ we find a candidate of discrete lattice model that may have annulus SLE$_4$ as its scaling limit. The local martingale for annulus SLE$_{8/3}$ is similar to those for chordal and radial SLE$_{8/3}$. But it seems that annulus SLE$_{8/3}$ does not satisfy the restriction property.
In this note, we prove a version of the conjectured duality of Schramm-Loewner Evolutions, by establishing exact identities in distribution between some boundary arcs of chordal $\SLE_κ$, $κ>4$, and appropriate versions of $\SLE_{\hatκ}$, $\hatκ=16/κ$.
Multiple Ising interfaces in annulus and $2N$-sided radial SLE
本文研究了环形区域临界Ising模型在稀有事件下的界面分布，证明了其收敛于$2N$边径向SLE$_3$，并通过对多重弦SLE的估计推广了格林函数的结论。
We give a simplified and complete proof of the convergence of the chordal exploration process in critical FK-Ising percolation to chordal SLE$_κ( κ-6)$ with $κ=16/3$. Our proof follows the classical excursion-construction of SLE$_κ(κ-6)$ processes in the continuum and we are thus led to introduce suitable cut-off stopping times in order to analyse the behaviour of the driving function of the discrete system when Dobrushin boundary condition collapses to a single point. Our proof is very different from [KS15, KS16] as it only relies on the convergence to the chordal SLE$_κ$ process in Dobrushin…
Suppose that D is a planar Jordan domain and x and y are distinct boundary points of D. Fix κ\in (4,8) and let η be an SLE_κprocess from x to y in D. We prove that the law of the time-reversal of ηis, up to reparameterization, an SLE_κprocess from y to x in D. More generally, we prove that SLE_κ(ρ_1;ρ_2) processes are reversible if and only if both ρ_i are at least κ/2-4, which is the critical threshold at or below which such curves are boundary filling. Our result supplies the missing ingredient needed to show that for all κ\in (4,8) the so-called conformal loop ensembles CLE_κ are canonicall…
The Schramm-Loewner evolution (SLE) can be simulated by dividing the time interval into N subintervals and approximating the random conformal map of the SLE by the composition of N random, but relatively simple, conformal maps. In the usual implementation the time required to compute a single point on the SLE curve is O(N). We give an algorithm for which the time to compute a single point is O(N^p) with p<1. Simulations with kappa=8/3 and kappa=6 both give a value of p of approximately 0.4.
We compute the almost-sure Hausdorff dimension of the double points of chordal SLE_kappa for kappa > 4, confirming a prediction of Duplantier-Saleur (1989) for the contours of the FK model. We also compute the dimension of the cut points of chordal SLE_kappa for kappa > 4 as well as analogous dimensions for the radial and whole-plane SLE_kappa(rho) processes for kappa > 0. We derive these facts as consequences of a more general result in which we compute the dimension of the intersection of two flow lines of the formal vector field e^{ih/chi}, where h is a Gaussian free field and chi > 0, of d…
The backward chordal Schramm-Loewner Evolution naturally defines a conformal welding homeomorphism of the real line. We show that this homeomorphism is invariant under the automorphism $x\mapsto -1/x$, and conclude that the associated solution to the welding problem (which is a natural renormalized limit of the finite time Loewner traces) is reversible. The proofs rely on an analysis of the action of analytic circle diffeomorphisms on the space of hulls, and on the coupling techniques of the second author.
We discuss the possible candidates for conformally invariant random non-self-crossing curves which begin and end on the boundary of a multiply connected planar domain, and which satisfy a Markovian-type property. We consider both, the case when the curve connects a boundary component to itself (chordal), and the case when the curve connects two different boundary components (bilateral). We establish appropriate extensions of Loewner's equation to multiply connected domains for the two cases. We show that a curve in the domain induces a motion on the boundary and that this motion is enough to f…
Given a simply connected planar domain D, distinct points x,y \in \partial D, and κ>0, the Schramm-Loewner evolution SLE_κis a random continuous non-self-crossing path in the closure of D from x to y. The SLE_κ(ρ_1;ρ_2) processes, defined for ρ_1, ρ_2 > -2, are in some sense the most natural generalizations of SLE_κ. When κ\leq 4, we prove that the law of the time-reversal of an \SLE_κ(ρ_1;ρ_2) from x to y is, up to parameterization, an SLE_κ(ρ_2;ρ_1) from y to x. This assumes that the "force points" used to define SLE_κ(ρ_1;ρ_2) are immediately to the left and right of the SLE seed. A general…
We consider a transient chordal SLE$_κ(ρ_1,\dots,ρ_m)$ curve $η$ in $\mathbb{H}$ from $w$ to $\infty$ with force points $ v_1> \cdots >v_m$ in $(-\infty,w^-]$, which intersects and is not boundary-filling on $(-\infty,v_m)$. The main result is that there is an atomless locally finite Borel measure $μ_η$ on $η\cap (-\infty,v_m]$ such that for any $v<v_m$, the $d$-dimensional Minkowski content of $η\cap [v,v_m]$ exists and equals $μ_η[v,v_m]$, where $d=\frac{(\sum ρ_j+4)(κ-4-2\sum ρ_j)}{2κ}$ is the Hausdorff dimension of $η\cap [v,v_m]$. In the case that all $ρ_j=0$, this measure agrees with the…
We establish existence and uniqueness for Gaussian free field flow lines started at {\em interior} points of a planar domain. We interpret these as rays of a random geometry with imaginary curvature and describe the way distinct rays intersect each other and the boundary. Previous works in this series treat rays started at {\em boundary} points and use Gaussian free field machinery to determine which chordal \SLE_κ(ρ_1; ρ_2) processes are time-reversible when κ< 8. Here we extend these results to whole-plane \SLE_κ(ρ) and establish continuity and transience of these paths. In particular, we ex…
Simmons and Cardy recently predicted a formula for the probability that the chordal SLE(8/3) path passes to the left of two points in the upper half-plane. In this paper we give a rigorous proof of their formula. Starting from this result, we derive explicit expressions for several natural connectivity functions for SLE(8/3) bubbles conditioned to be of macroscopic size. By passing to a limit with such a bubble we construct a certain chordal restriction measure and in this way obtain a proof of a formula for the probability that two given points are between two commuting SLE(8/3) paths. The on…
We characterize and describe all random subsets $K$ of a given simply connected planar domain (the upper half-plane $\H$, say) which satisfy the ``conformal restriction'' property, i.e., $K$ connects two fixed boundary points (0 and $\infty$, say) and the law of $K$ conditioned to remain in a simply connected open subset $D$ of $\H$ is identical to that of $Φ(K)$, where $Φ$ is a conformal map from $\H$ onto $D$ with $Φ(0)=0$ and $Φ(\infty)=\infty$. The construction of this family relies on the stochastic Loewner evolution (SLE) processes with parameter $κ\le 8/3$ and on their distortion under …
We resolve a conjecture of Sheffield that $\SLE(4)$, a conformally invariant random curve, is the universal limit of the chordal zero-height contours of random surfaces with isotropic, uniformly convex potentials. Specifically, we study the \emph{Ginzburg-Landau $\nabla ϕ$ interface model} or \emph{anharmonic crystal} on $D_n = D \cap \tfrac{1}{n} \Z^2$ for $D \subseteq \C$ a bounded, simply connected Jordan domain with smooth boundary. This is the massless field with Hamiltonian $\CH(h) = \sum_{x \sim y} \CV(h(x) - h(y))$ with $\CV$ symmetric and uniformly convex and $h(x) = ϕ(x)$ for $x \in …
Massive SLE$_4$ and the scaling limit of the massive harmonic explorer
本文完整严格证明了有质量调和探索子的标度极限为共形协变、相对弦向SLE$_4$绝对连续的massive SLE$_4$，还证明合适边界条件下massive SLE$_4$可作为有质量高斯自由场的等值线耦合。
We define multiple chordal SLEs in a simply connected domain by considering a natural configurational measure on paths. We show how to construct these measures so that they are conformally covariant and satisfy certain boundary perturbation and Markov properties, as well as a cascade relation. As an example of our construction, we derive the scaling limit of Fomin's identity in the case of two paths directly; that is, we prove that the probability that an SLE(2) and a Brownian excursion do not intersect can be given in terms of the determinant of the excursion hitting matrix. Finally, we defin…
Developing the theory of two-sided radial and chordal $\mathit{SLE}$, we prove that the natural parametrization on $\mathit{SLE}_κ$ curves is well defined for all $κ<8$. Our proof uses a two-interior-point local martingale.
Multiple chordal SLE($κ$) and quantum Calogero-Moser system
本文研究带标记边界点的多重弦SLE($κ$)，利用共形场论Coulomb气体方法构造两类配分函数解，并证明其对应量子Calogero–Moser本征态。
We consider the measure on multiple chordal Schramm-Loewner evolution ($SLE_κ$) curves. We establish a derivative estimate and use it to give a direct proof that the partition function is $C^2$ if $κ<4$.
We prove an upper bound on the optimal Hölder exponent for the chordal SLE path parameterized by capacity and thereby establish the optimal exponent as conjectured by J. Lind. We also give a new proof of the lower bound. Our proofs are based on the sharp estimates of moments of the derivative of the inverse map. In particular, we improve an estimate of the second author.
Large deviations of SLE(0+) variants in the capacity parameterization
本文证明了全弦状、径向和多弦状SLE(0+)曲线在容量参数化下的大偏差原理，通过Loewner能量的变体作为速率函数，并改进了拓扑结构和径向情况的处理方法。
We revisit the convergence of loop-erased random walk, LERW, to SLE(2) when the curves are parametrized by capacity. We construct a coupling of the chordal version of LERW and chordal SLE(2) based on the Green's function for LERW as martingale observable and using an elementary discrete-time Loewner "difference" equation. This coupling is different than the ones previously considered in this context. Our recent work (arXiv:1603.05203) on the convergence of LERW parametrized by length to SLE(2) parameterized by Minkowski content uses specific features of the coupling constructed here.
We study the probability that chordal $\text{SLE}_{8/3}$ in the unit disk from $\exp(ix)$ to 1 avoids the disk of radius $q$ centered at zero. We find the initial/boundary-value problem satisfied by this probability as a function of $x$ and $a=\ln q$, and show that asymptotically as $q$ tends to one this probability decays like $\exp(-cx/(1-q))$ with $c=5π/8$ for $x\in[0,π]$. We also give a representation of this probability as a functional of a Legendre process.
We review some of the results related to conformal restriction: the chordal case and the radial case. We describe Brownian intersection exponents, conformal restriction property and SLE, and study their properties.
We study the probabilities with which chordal Schramm-Loewner Evolutions (SLE) visit small neighborhoods of boundary points. We find formulas for general chordal SLE boundary visiting probability amplitudes, also known as SLE boundary zig-zags or order refined SLE multi-point Green's functions on the boundary. Remarkably, an exact answer can be found to this important SLE question for an arbitrarily large number of marked points. The main technique employed is a spin chain-Coulomb gas correspondence between tensor product representations of a quantum group and functions given by Dotsenko-Fatee…
We present a mathematical proof of theoretical predictions made by Arguin and Saint-Aubin, as well as by Bauer, Bernard, and Kytola, about certain non-local observables for the two-dimensional Ising model at criticality by combining Smirnov's recent proof of the fact that the scaling limit of critical Ising interfaces can be described by chordal SLE(3) with Kozdron and Lawler's configurational measure on mutually avoiding chordal SLE paths. As an extension of this result, we also compute the probability that an SLE(k) path, k in (0,4], and a Brownian motion excursion do not intersect.
Classification of commutation relation for multi-chordal SLE
本文对多弦 SLE 的 BPZ 方程线性解空间进行分类，通过三角坐标变换化为有限维线性代数，确定了平移不变解空间的维数，恢复了 Catalan 维数公式，并构造了高斯自由场水平线的配分函数。
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