Hao Wu
We derive the alternating arm exponents of critical Ising model. We obtain six different patterns of alternating boundary arm exponents which correspond to the boundary conditions $(\ominus\oplus)$, $(\ominus\text{free})$ and $(\text{free}\text{free})$, and the alternating interior arm exponents.
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Radial BPZ equations and partition functions of FK-Ising interfaces conditional on one-arm event
本文研究径向设置下Dubédat交换关系导出的径向BPZ方程，构造其正解，并证明多边形中FK-Ising界面在一臂事件条件下的配分函数构成该方程的正解。
We derive boundary arm exponents and interior arm exponents for SLE$(κ)$. Combining with the possible convergence of critical lattice models to SLE, these exponents would give the corresponding alternating half-plane arm exponents and alternating plane arm exponents for the lattice models.
Basic techniques in two-dimensional critical Ising percolation with investigation of scaling relations
本文研究了二维正方晶格上高温Ising模型在临界外场下的渗流问题。我们发现，所有的标度关系，除了一个单一的超标度关系，举行的幂律假设的单臂路径和四臂路径。
We consider the ferromagnetic nearest-neighbor Ising model on regular trees (Bethe lattice), which is well-known to undergo a phase transition in the absence of an external magnetic field. The behavior of the model at critical temperature can be described in terms of various critical exponents; one of them is the critical 1-arm exponent $ρ$, which characterizes the rate of decay of the (root) magnetization. The crucial quantity we analyze in this work is the thermal expectation of the root spin on a finite subtree, where the expected value is taken with respect to a probability measure related…
Schramm Loewner Evolution (SLE) is a one-parameter family of random planar curves introduced by Oded Schramm in 1999 as the candidates for the scaling limits of the interfaces in the planar critical lattice models. This is the only possible process with conformal invariance and a certain "domain Markov property". In 2010, Chelkak and Smirnov proved the conformal invariance of the scaling limits of the critial planar FK-Ising model which gave the convergence of the interface to SLE$_{16/3}$. We derive the arm exponents of SLE$_κ$ for $κ\in (4,8)$. Combining with the convergence of the interface…
We derive boundary arm exponents for SLE. Combining with the convergence of critical lattice models to SLE, these exponents would give the alternating half-plane arm exponents for the corresponding lattice models.
The 1-arm exponent $ρ$ for the ferromagnetic Ising model on $\mathbb{Z}^d$ is the critical exponent that describes how fast the critical 1-spin expectation at the center of the ball of radius $r$ surrounded by plus spins decays in powers of $r$. Suppose that the spin-spin coupling $J$ is translation-invariant, $\mathbb{Z}^d$-symmetric and finite-range. Using the random-current representation and assuming the anomalous dimension $η=0$, we show that the optimal mean-field bound $ρ\le1$ holds for all dimensions $d>4$. This significantly improves a bound previously obtained by a hyperscaling inequ…
One-arm exponents of the high-dimensional Ising model
研究高维伊辛模型中一臂概率的衰减规律，揭示其上临界维度为6，推翻传统伊辛模型的结论。
Connection probabilities and RSW-type bounds for the FK Ising model
我们证明了Russo-Seymour-Welsh型的交叉概率的一致边界的FK伊辛模型在临界状态，独立的边界条件。我们的证明主要依赖于Smirnov的费米子观测的FK伊辛模型，这使我们能够得到精确的估计边界连接的概率。它仍然是纯离散的，特别是我们不使用任何连续极限，它可以用来直接推导出几个值得注意的结果-一些新的和一些不-其中的事实，即没有自发磁化临界，紧密性的接口，并存在几个临界指数，特别是半平面单臂指数。
We consider the planar Ising model in rectangle $(Ω; x^L, x^R, y^R, y^L)$ with alternating boundary condition: $\ominus$ along $(x^Lx^R)$ and $(y^Ry^L)$, $ξ^R\in\{\oplus, \text{free}\}$ along $(x^Ry^R)$, and $ξ^L\in\{\oplus, \text{free}\}$ along $(y^Lx^L)$. We prove that the interface of critical Ising model with these boundary conditions converges to the so-called hypergeometric SLE$_3$. The method developed in this paper does not require constructing new holomorphic observable and the input is the convergence of the interface with Dobrushin boundary condition. This method could be applied to…
Comparison of arm exponents in planar FK-percolation
该研究针对平面FK渗流的交替双臂事件概率不等式问题，在连续相变区间内通过多项式因子改进了临界FK渗流的FKG不等式，证明了交替双臂指数严格大于两个单臂指数之和，回应了相关领域的公开问题。
A CLT for winding angles of the arms for critical planar percolation
考虑二维临界渗流。在环A（l，n）上有k条不相交的黑白相间的臂交叉的条件下，证明了一个中心极限定理和臂缠绕角的方差估计（当n个\rightarrow \infty，l固定时）.这一结果证实了Beffara和Nolin（Ann. Probab. 39：1286- 1304，2011）。利用这个定理，我们还得到了多臂初始无限簇（IIC）测度的CLT。
Crossing probabilities in topological rectangles for the critical planar FK-Ising model
我们认为在临界的FK伊辛模型在两个维度。本文得到了任意拓扑矩形的交叉概率的界，它是关于边界条件一致的，推广了[DCHN 11]和[CS 12]的结果。我们的结果依赖于新的离散复分析技术，介绍了在[车12]。我们详细介绍了一些应用程序，特别是所谓的普适指数的计算，准乘法性质的手臂概率的证明，并跨越概率的经典伊辛模型的界限。
The fuzzy Potts model in the plane: Scaling limits and arm exponents
该研究针对平面临界FK渗流染色得到的模糊Potts模型，证明其在共形不变标度极限假设下收敛于共形环系综变体，且离散与连续模型臂指数一致，进而确定了该模型的臂指数。
This paper is studying the critical regime of the planar random-cluster model on $\mathbb Z^2$ with cluster-weight $q\in[1,4)$. More precisely, we prove crossing estimates in quads which are uniform in their boundary conditions and depend only on their extremal lengths. They imply in particular that any fractal boundary is touched by macroscopic clusters, uniformly in its roughness or the configuration on said boundary. Additionally, they imply that any sub-sequential scaling limit of the collection of interfaces between primal and dual clusters is made of loops that are non-simple. We also ob…
Polynomial rate of relaxation for the Glauber dynamics of infinite-volume critical Ising model
本文研究任意维度无限体积临界铁磁 Ising 模型的 Glauber 动力学，在有限体积 log-Sobolev 常数和 1-臂指数假设下，证明等位置时间自旋相关函数随时间多项式衰减。
We consider the Ising model at its critical temperature with external magnetic field $ha^{15/8}$ on $a\mathbb{Z}^2$. We give a purely probabilistic proof, using FK methods rather than reflection positivity, that for $a=1$, the correlation length is $\geq const.~h^{-8/15}$ as $h\downarrow0$. We extend to the $a\downarrow0$ continuum limit the FK-Ising coupling for all $h>0$, and obtain tail estimates for the largest renormalized cluster area in a finite domain as well as an upper bound with exponent $1/8$ for the one-arm event. Finally, we show that for $a=1$, the average magnetization, $\mathc…
One-arm exponents of high-dimensional percolation revisited
本文针对高维(d>6)渗流，提出简短证明，统一处理全空间与半空间单臂概率的常数阶估计，改进熵技术并利用关联长度尖锐估计，受高维Ising模型相关工作启发。
Smirnov's observable for free boundary conditions, interfaces and crossing probabilities
我们证明了在自由边界条件的存在下，在临界伊辛模型的Smirnov的费米子观测变量的收敛结果。我们的分析的一个应用是洪勒和基特勒关于临界伊辛界面在正负自由边界条件下收敛于偶极SLE（3）的定理的简单证明，以及这个结果在任意数量的带有正负或自由边界条件的弧上的推广。另一个应用是计算具有任意数量的交替有线/自由边界弧的FK-伊辛模型中交叉概率的标度极限。我们还推导了自旋伊辛模型的一个新的交叉公式。
A matrix is called totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative). Consider the Ising model with free boundary conditions and no external field on a planar graph $G$. Let $a_1,\dots,a_k,b_k,\dots,b_1$ be vertices placed in a counterclockwise order on the outer face of $G$. We show that the $k\times k$ matrix of the two-point spin correlation functions \[ M_{i,j} = \langle σ_{a_i} σ_{b_j} \rangle \] is totally nonnegative. Moreover, $\det M > 0$ if and only if there exist $k$ pairwise vertex-disjoint paths that connect $a_i$ with $b_i$. We also c…
We prove convergence of the 2- and 4-point fermionic observables of the FK-Ising model on simply connected domains discretised by a planar isoradial lattice in massive (near-critical) scaling limit. The former is alternatively known as a (fermionic) martingale observable (MO) for the massive interface, and in particular encapsulates boundary visit probabilties of the interface. The latter encodes connection probabilities in the 4-point alternating (generalised Dobrushin) boundary condition, whose exact convergence is then further analysed to yield crossing estimates for general boundary condit…
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。
Bond percolation on isoradial graphs: criticality and universality
在等径图上的渗流研究中，我们证明了正则键渗流在可平面图的等径嵌入上的临界性，从而推广了较早的关于均匀和非均匀正方形、三角形和其它格点的著名结果。这是通过星三角形变换实现的，通过在等径图族中传输盒交叉属性。因此，我们得到的普遍性，这些模型在临界点，在这个意义上说，单臂和2 j-交替臂临界指数（因此也是连通性和体积指数）是恒定的家庭这样的渗流过程。所讨论的等径图是那些在嵌入和迹系上满足一定弱条件的图。这类图包括，例如，周期图的等距嵌入，以及从菱形彭罗斯镶嵌导出的图。
Universality for bond percolation in two dimensions
正方形、三角形和六边形晶格上的所有（非）均质键渗流模型都属于同一普适类，因为它们在临界点具有相同的临界指数（假设指数存在）。这是证明使用星三角形变换和盒交叉性质。所讨论的指数是单臂指数$ρ$、$j\ge1$的2 j-交替臂指数$ρ_{2j}$、体积指数$δ$和连通性指数$η$。根据Kesten早期的结果，这意味着对于满足某种附加假设的任何模型，如这三种晶格上的均匀键渗流模型，近临界指数$β$、$γ$、$ν$、$Δ$（假设这些指数存在）也具有普适性。
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