Gregory F. Lawler, Wendelin Werner
We define a natural conformally invariant measure on unrooted Brownian loops in the plane and study some of its properties. We relate this measure to a measure on loops rooted at a boundary point of a domain and show how this relation gives a way to ``chronologically add Brownian loops'' to simple curves in the plane.
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Non-existence of several random fractals in Brownian motion and Brownian loop soup
本文通过统一方法证明了三种随机分形的不存在性，包括平面布朗运动的先驱三重点、平面和三维布朗运动的先驱双断点，以及临界强度下平面布朗循环汤簇边界上的双点，解决了相关开放问题。
Generalized intersection exponents and local cut points for three-dimensional Brownian loop soup
本文研究了三维布朗运动循环汤在亚临界强度下的广义非交集概率，推导了广义交集指数的存在性，并通过定制的分离引理证明了这些概率的估计。同时，将三维布朗运动循环汤局部割点集的Hausdorff维度与广义交集指数联系起来，证明在强度为零时广义交集指数连续，并退化为经典的布朗运动交集指数。特别地，这表明对于足够小的强度参数，局部割点集的Hausdorff维度严格大于1。
The real layering field of Brownian loop soup and the Gaussian multiplicative chaos
本文研究复平面有界单连通域中布朗环汤的分层场，证明其经重整化后收敛到亚临界高斯乘性混沌，主要技术为维纳-伊藤混沌展开，并计算了$n$点函数及其共形协变性和边界行为。
The random walk loop soup is a Poissonian ensemble of lattice loops; it has been extensively studied because of its connections to the discrete Gaussian free field, but was originally introduced by Lawler and Trujillo Ferreras as a discrete version of the Brownian loop soup of Lawler and Werner, a conformally invariant Poissonian ensemble of planar loops with deep connections to conformal loop ensembles (CLEs) and the Schramm-Loewner evolution (SLE). Lawler and Trujillo Ferreras showed that, roughly speaking, in the continuum scaling limit, ``large'' lattice loops from the random walk loop sou…
The Brownian loop soup introduced in Lawler and Werner (2004) is a Poissonian realization from a sigma-finite measure on unrooted loops. This measure satisfies both conformal invariance and a restriction property. In this paper, we define a random walk loop soup and show that it converges to the Brownian loop soup. In fact, we give a strong approximation result making use of the strong approximation result of Komlós, Major, and Tusnády. To make the paper self-contained, we include a proof of the approximation result that we need.
Coupling Brownian loop soups and random walk loop soups at all polynomial scales
本文研究\(\mathbb{Z}^d\)上布朗环汤与随机游走环汤的耦合，找到去除时间长度限制的方法，使耦合适用于所有多项式尺度，还建立了离散和连续时间随机游走环汤耦合及相关KMT耦合。
Multi-window trace connectivity in subcritical planar Brownian loop soups
研究亚临界平面布朗环汤中相交环轨迹簇与固定数量分离收缩体圆盘相交概率，通过多目标布朗环测度估计等方法，得出该概率在对数指数有损失时是单臂概率乘积的结论，部分成果源于前人。
We construct loop soups for general Markov processes without transition densities and show that the associated permanental process is equal in distribution to the loop soup local time. This is used to establish isomorphism theorems connecting the local time of the original process with the associated permanental process. Further properties of the loop measure are studied.
We introduce a natural "massive" version of the Brownian loop soup of Lawler and Werner which displays conformal covariance and exponential decay. We show that this massive Brownian loop soup arises as the near-critical scaling limit of a random walk loop soup with killing and is related to the massive SLE(2) identified by Makarov and Smirnov as the near-critical scaling limit of a loop-erased random walk with killing. We conjecture that the massive Brownian loop soup describes the zero level lines of the massive Gaussian free field, and discuss possible relations to other models, such as Isin…
We describe simple properties of some soups of unoriented Markov loops and of some soups of oriented Markov loops that can be interpreted as a spatial Markov property of these loop-soups. This property of the latter soup is related to well-known features of the uniform spanning trees (such as Wilson's algorithm) while the Markov property of the former soup is related to the Gaussian Free Field and to identities used in the foundational papers of Symanzik, Nelson, and of Brydges, Fröhlich and Spencer or Dynkin, or more recently by Le Jan.
Universality for the 2D Random Walk Loop Soup
研究二维平面图上随机游走环汤的标度极限，证明其收敛于布朗环汤，基于随机游走收敛、Russo-Seymour-Welsh估计和图的有界密度，利用Wilson算法和Schramm的有限性定理。
We study the structure of Brownian loop-soup clusters in two dimensions. Among other things, we obtain the following decomposition of the clusters with critical intensity: When one conditions a loop-soup cluster by its outer boundary $γ$ (which is known to be an SLE(4)-type loop), then the union of all excursions away from $γ$ by all the Brownian loops in the loop-soup that touch $γ$ is distributed exactly like the union of all excursions of a Poisson point process of Brownian excursions in the domain enclosed by $γ$. A related result that we derive and use is that the couplings of the Gaussia…
We consider the loop soup at intensity ${1\over 2}$ conditioned on having local time $0$ on a set of vertices with positive occupation field in their vicinities. We give a relation between this loop soup and the usual loop soup conditioned on its local times. We deduce a domain Markov property for the loop soup, in the vein of the discrete Markov property proved by Werner: when exploring a cluster, the bridges outside the cluster form a Poisson point process. We show how it is related to the property due to Le Jan that the local times of the loop soup are distributed as the squares of a Gaussi…
We consider the random walk loop soup on the discrete half-plane corresponding to a central charge c in (0, 1]. We look at the clusters of discrete loops and show that the scaling limit of the outer boundaries of outermost clusters is the CLE(kappa) loop ensemble, with the same relation between kappa and c as in the continuum Brownian setting.
A Study in Markov Chains, Loop-Erased Random Walk and Loop Soups
本文探讨了马尔可夫链、擦除环路随机游走及环路汤的相关概念，定义了带“杀灭”的暂态与常返链、格林函数等，揭示了LERW与UST的关系，并引入环路测度与“生长环路”模型。
On the gap between cluster dimensions of loop soups on $\mathbb{R}^3$ and the metric graph of $\mathbb{Z}^3$
本文研究了三维空间中循环汤簇聚类维度与三维整格度量图簇维度之间的差距，发现度量图簇的尺度极限严格大于连续域循环汤簇。
We define and study a set of operators that compute statistical properties of the Brownian Loop Soup, a conformally invariant gas of random Brownian loops (Brownian paths constrained to begin and end at the same point) in two dimensions. We prove that the correlation functions of these operators have many of the properties of conformal primaries in a conformal field theory, and compute their conformal dimension. The dimensions are real and positive, but have the novel feature that they vary continuously as a periodic function of a real parameter. We comment on the relation of the Brownian Loop…
Boundary operators in the Brownian loop soup
该研究通过共形块展开揭示了布朗运动环汤中亚临界相的无限多个边界算符，并解释了其物理意义。
Singularity of the loops within a cable-graph loop-soup conditioned by its occupation time
本文研究了在电缆图上布朗运动循环汤在总占用时间场条件下的奇异性质，揭示了循环分布的奇异行为及其成因。
We initiate the study on chemical distances of percolation clusters for level sets of two-dimensional discrete Gaussian free fields as well as loop clusters generated by two-dimensional random walk loop soups. One of our results states that the chemical distance between two macroscopic annuli away from the boundary for the random walk loop soup at the critical intensity is of dimension 1 with positive probability. Our proof method is based on an interesting combination of a theorem of Makarov, isomorphism theory and an entropic repulsion estimate for Gaussian free fields in the presence of a h…
Percolation threshold for metric graph loop soup
本文证明了一大类暂态度量图（包括 $\mathbb{Z}^d$ 等）上度量图环路汤发生渗流的临界阈值为 $1/2$。
We make a few elementary observations that relate directly the items mentioned in the title. In particular, we note that when one superimposes the random current model related to the Ising model with an independent Bernoulli percolation model with well-chosen weights, one obtains exactly the FK-percolation (or random cluster model) associated with the Ising model. We also point out that this relation can be interpreted via loop-soups, combining the description of the sign of a Gaussian Free Field on a discrete graph knowing its square (and the relation of this question with the FK-Ising model)…
We show that supersymmetric (susy) hyperbolic isomorphism theorems that relate Vertex Reinforced Jump Processes and $H^{2|2}$ field, introduced in [2] and [3], are annealed version of isomorphism theorems relating Markov processes and Gaussian free field, with the help of a Bayes formula that relates susy hyperbolic field to susy free field. On the other hand, we also prove a BFS-Dynkin's isomorphism theorem for reinforced loop soup. Moreover, we provide yet another proof of BFS-Dynkin's isomorphism for VRJP a la Feynman-Kac.
Intensity doubling for Brownian loop-soups in high dimensions
研究高维空间中布朗运动环汤的强度倍增现象，通过分解环簇发现两种分布相同的家族，揭示了大规模循环在标度极限下的行为。
Lupu introduced a coupling between a random walk loop-soup and a Gaussian free field, where the sign of the field is constant on each cluster of loops. This coupling is a signed version of isomorphism theorems relating the square of the GFF to the occupation field of Markovian trajectories. His construction starts with a loop-soup, and by adding additional randomness samples a GFF out of it. In this article we provide the inverse construction: starting from a signed free field and using a self-interacting random walk related to this field, we construct a random walk loop-soup. Our construction…
Several stochastic processes related to transient Lévy processes with potential densities $u(x,y)=u(y-x)$, that need not be symmetric nor bounded on the diagonal, are defined and studied. They are real valued processes on a space of measures $V$ endowed with a metric $d$. Sufficient conditions are obtained for the continuity of these processes on $(V,d)$. The processes include $n$-fold self-intersection local times of transient Lévy processes and permanental chaoses, which are `loop soup $n$-fold self-intersection local times' constructed from the loop soup of the Lévy process. Loop soups are …
Parity questions in critical planar Brownian loop-soups (or "where did the free planar bosons go?")
该研究探讨临界二维布朗运动环汤的占据时间场与高斯自由场平方的关联，证明场无法确定环的访问点集，且稠密特殊点属环汤的条件概率为1/2，环分解也分两类各占1/2概率。
We compute analytically and in closed form the four-point correlation function in the plane, and the two-point correlation function in the upper half-plane, of layering vertex operators in the two dimensional conformally invariant system known as the Brownian Loop Soup. These correlation functions depend on multiple continuous parameters: the insertion points of the operators, the intensity of the soup, and the charges of the operators. In the case of the four-point function there is non-trivial dependence on five continuous parameters: the cross-ratio, the intensity, and three real charges. T…
Scaling Limit of Critical Loop Soup Clusters in Three, Four, and Five Dimensions
本文证明了在三维、四维和五维整数格上，临界环汤簇（等价于临界高斯自由场水平集簇）存在标度极限，为三维逾渗模型提供了首个标度极限结果。
We show that if one conditions a cluster in a Brownian loop-soup $L$ (of any intensity) in a two-dimensional domain by a portion $l$ of its outer boundary, then in the remaining domain, the union of all the loops of $L$ that touch $l$ satisfies the conformal restriction property while the other loops in $L$ form an independent loop-soup. This result holds when one discovers $l$ in a natural Markovian way, such as in the exploration procedures that have been defined in order to actually construct the Conformal Loop Ensembles as outer boundaries of loop-soup clusters. This result implies among o…
Multiplicative chaos of the Brownian loop soup
本文构造了布朗环汤厚点上的乘法混沌测度，证明在临界强度下其与高斯自由场的双曲余弦一致，从而揭示布朗乘法混沌、高斯自由场与刘维尔测度间的深刻联系，并发现精确可积性。
Formation of infinite loops for an interacting bosonic loop soup
本文计算了非平均场能量下玻色气体Feynman环表示的极限测度，证明了高密度下随机交织的正权重指示了玻色-爱因斯坦凝聚，并发现临界密度发生偏移且密度在临界点存在跳跃间断。
It has been recently understood (arXiv:1212.2885, arXiv:1310.4764, arXiv:1410.0605) that for a general class of percolation models on $\mathbb{Z}^d$ satisfying suitable decoupling inequalities, which includes i.a.\ Bernoulli percolation, random interlacements and level sets of the Gaussian free field, large scale geometry of the unique infinite cluster in strongly percolative regime is qualitatively the same; in particular, the random walk on the infinite cluster satisfies the quenched invariance principle, Gaussian heat-kernel bounds and local CLT. In this paper we consider the random walk lo…
Emergence of interlacements from the finite volume Bose soup
本文证明当粒子密度超过临界值时，有限体积玻色环汤收敛为全空间玻色环汤与随机交织泊松点过程的叠加，强度由过剩密度决定，并涵盖自由与平均场两种情形。
The Brownian loop soup (BLS) is a conformally invariant statistical ensemble of random loops in two dimensions characterized by an intensity $λ>0$. Recently, we constructed families of operators in the BLS and showed that they transform as conformal primary operators. In this paper we provide an explicit expression for the BLS stress-energy tensor and compute its operator product expansion with other operators. Our results are consistent with the conformal Ward identities and our previous result that the central charge is $c = 2 λ$. In the case of domains with boundary we identify a boundary o…
We study some properties of a class of random connected planar fractal sets induced by a Poissonian scale-invariant and translation-invariant point process. Using the second-moment method, we show that their Hausdorff dimensions are deterministic and equal to their expectation dimension. We also estimate their low-intensity limiting behavior. This applies in particular to the "conformal loop ensembles" defined via Poissonian clouds of Brownian loops for which the expectation dimension has been computed by Schramm, Sheffield and Wilson.
Three-dimensional Brownian loop soup clusters
该研究探讨了三维空间中布朗运动环汤簇的相变现象，证明了在高强度下所有环连接成单一簇，并展示了环测度的守恒性。
Multiple points on the boundaries of Brownian loop-soup clusters
本文研究单位圆盘内布朗环汤簇边界上的多重点，证明简单点和二重点的豪斯多夫维数分别为 $2-ξ_c(2)$ 和 $2-ξ_c(4)$，并建立分离引理以定义广义相交指数 $ξ_c(k, λ)$，揭示其与断开指数 $ξ_c(k)$ 的极限关系。
Crossing exponent in the Brownian loop soup
本文研究二维有界区域内亚临界布朗环汤的环簇性质，推导了环带跨越渐近概率、圆盘击中概率衰减速率，刻画了簇极集，并揭示一、二维布朗环汤的关联及新的临界强度相变点。
We study vertex-like operators built from the Brownian loop soup in the limit as the loop soup intensity tends to infinity. More precisely, following Camia, Gandolfi and Kleban (Nuclear Physics B 902, 2016), we take a Brownian loop soup in a planar domain and assign a random sign to each loop. We then consider random fields defined by taking, at every point of the domain, the exponential of a purely imaginary constant times the sum of the signs associated to the loops that wind around that point. As smaller loops are included in the count, that sum diverges logarithmically with the diameter of…
We show the result that is stated in the title of the paper, which has consequences about decomposition of Brownian loop-soup clusters in two dimensions.
We define a large new class of conformal primary operators in the ensemble of Brownian loops in two dimensions known as the ``Brownian loop soup,'' and compute their correlation functions analytically and in closed form. The loop soup is a conformally invariant statistical ensemble with central charge $c = 2 λ$, where $λ> 0$ is the intensity of the soup. Previous work identified exponentials of the layering operator $e^{i βN(z)}$ as primary operators. Each Brownian loop was assigned $\pm 1$ randomly, and $N(z)$ was defined to be the sum of these numbers over all loops that encircle the point $…
(Non-)coincidence of critical parameters for Poisson Zoos and Loop Soup Percolation on $\mathbb{Z^d}$, $d > 4$, and $\mathbb{T}_d$,$ d \ge 3$
本研究针对d>4的Z^d及d≥3的T_d，探讨泊松动物园与环汤渗流的临界参数（非）重合性，证明d≥5时Z^d上离散图与对应度量图的环汤渗流临界参数存在差异，还推导了树上泊松动物园渗流与磁化率的临界参数重合结论，并发展了长程相关渗流对伯努利增强的敏感性概念。
We study spin systems defined by the winding of a random walk loop soup. For a particular choice of loop soup intensity, we show that the corresponding spin system is reflection-positive and is dual, in the Kramers-Wannier sense, to the spin system $\text{sgn}(φ)$ where $φ$ is a discrete Gaussian free field. In general, we show that the spin correlation functions have conformally covariant scaling limits corresponding to the one-parameter family of functions studied by Camia, Gandolfi and Kleban (Nuclear Physics B, 902, 2016) and defined in terms of the winding of the Brownian loop soup. These…
Cover times of the massive random walk loop soup
研究二维大质量随机游动环路汤对 ${\mathbb Z}^2$ 子集的覆盖时间分布极限，允许消亡率依赖集合大小，证明在接近极端情形下覆盖时间服从 Gumbel 分布。
A permanental field, $ψ=\{ψ(ν),ν\in {\mathcal{V}}\}$, is a particular stochastic process indexed by a space of measures on a set $S$. It is determined by a kernel $u(x,y)$, $x,y\in S$, that need not be symmetric and is allowed to be infinite on the diagonal. We show that these fields exist when $u(x,y)$ is a potential density of a transient Markov process $X$ in $S$. A permanental field $ψ$ can be realized as the limit of a renormalized sum of continuous additive functionals determined by a loop soup of $X$, which we carefully construct. A Dynkin-type isomorphism theorem is obtained that relat…
A switching identity for cable-graph loop soups and Gaussian free fields
本文提出电缆图环路汤与高斯自由场的切换恒等式，证明两点同簇条件等价于添加随机奇数条独立布朗游程，并由此分析无限图上符号簇的大尺度行为。
This article deals with limit theorems for certain loop variables for loop soups whose intensity approaches infinity. We first consider random walk loop soups on finite graphs and obtain a central limit theorem when the loop variable is the sum over all loops of the integral of each loop against a given one-form on the graph. An extension of this result to the noncommutative case of loop holonomies is also discussed. As an application of the first result, we derive a central limit theorem for windings of loops around the faces of a planar graphs. More precisely, we show that the winding field …
The two-dimensional Brownian loop-soup is a Poissonian random collection of loops in a planar domain with an intensity parameter c. When c is not greater than 1, we show that the outer boundaries of the loop clusters are disjoint simple loops (when c>1, there is almost surely only one cluster) that satisfy certain conformal restriction axioms. We prove various results about loop-soups, cluster sizes, and the c=1 phase transition. Combining this with the results of another paper of ours on the Markovian characterization of simple conformal loop ensembles (CLE), this proves that these outer boun…
We are interested in path decompositions of a perturbed reflecting Brownian motion (PRBM) at the hitting times and at the minimum. Our study relies on the loop soups developed by Lawler and Werner [10] and Le Jan [13]-[14], in particular on a result discovered by Lupu [15] identifying the law of the excursions of the PRBM above its past minimum with the loop measure of Brownian bridges.
A note on $α$-permanent and loop soup
本文揭示了代数中α-永久与概率中环路汤之间的深刻联系，并对由*-森林相关矩阵构成的块矩阵给出了α-永久的显式展开，同时提供了组合与概率两种证明。
Percolation of discrete GFF in dimension two I. Arm events in the random walk loop soup
本文研究了二维随机游走路环汤，估计了次临界和临界状态下两个大连通环路分量相互靠近的概率，并基于随机游走分离与环路手术开发了局部性和准可乘性等工具，为后续研究离散高斯自由场水平集连通性奠定基础。
Loop soup percolation and boundary dimension on metric trees
该研究确定了具有适当电缆度量的暂态局部有限加权树上布朗环汤的渗流阈值，给出了伯努利渗流 Lyons 公式的类似物，并将分类扩展至超临界 Galton-Watson 树及相关实树。
Reinforced Loop Soup via Wilson's Algorithm
本文通过改进的威尔逊算法构建了强化环汤，并建立了其与标准马尔可夫同构定理的联系。
For random collections of self-avoiding loops in two-dimensional domains, we define a simple and natural conformal restriction property that is conjecturally satisfied by the scaling limits of interfaces in models from statistical physics. This property is basically the combination of conformal invariance and the locality of the interaction in the model. Unlike the Markov property that Schramm used to characterize SLE curves (which involves conditioning on partially generated interfaces up to arbitrary stopping times), this property only involves conditioning on entire loops and thus appears a…
We provide a decomposition of the trace of the Brownian motion into a simple path and an independent Brownian soup of loops that intersect the simple path. More precisely, we prove that any subsequential scaling limit of the loop erased random walk is a simple path (a new result in three dimensions), which can be taken as the simple path of the decomposition. In three dimensions, we also prove that the Hausdorff dimension of any such subsequential scaling limit lies in $(1,\frac53]$. We conjecture that our decomposition characterizes uniquely the law of the simple path. If so, our results woul…
The Brownian loop-catcher
研究引入布朗环捕手，它在LERW和布朗轨迹间插值，为特定中心电荷的布朗环汤簇提供规范延续，给出其恢复性质、外边界特征等，通过有限图上随机游走环捕手经格点近似收敛得到，还提出格林函数测试并扩展到三维。
We explore the geometric meaning of the so-called zeta-regularized determinant of the Laplace-Beltrami operator on a compact surface, with or without boundary. We relate the $(-c/2)$-th power of the determinant of the Laplacian to the appropriately regularized partition function of a Brownian loop soup of intensity $c$ on the surface. This means that, in a certain sense, decorating a random surface by a Brownian loop soup of intensity $c$ corresponds to weighting the law of the surface by the $(-c/2)$-th power of the determinant of the Laplacian. Next, we introduce a method of regularizing a L…
The Brownian loop soup is a conformally invariant statistical ensemble of random loops in two dimensions characterized by an intensity $λ>0$, with central charge $c=2 λ$. Recent progress resulted in an analytic form for the four-point function of a class of scalar conformal primary "layering vertex operators" $\mathcal{O}_β$ with dimensions $(Δ, Δ)$, with $Δ= \fracλ{10}(1-\cosβ)$, that compute certain statistical properties of the model. The Virasoro conformal block expansion of the four-point function revealed the existence of a new set of operators with dimensions $(Δ+ k/3, Δ+ k'/3)$, for al…
Arm events in critical planar loop soups
本文研究二维临界 Brownian 环汤的体区四臂与半平面两臂事件，借助其与 Gaussian free field 的耦合和 Lupu–Werner 估计建立常数倍概率界，并推出随机游走环汤上界及相关场穿越结论。
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