Stanislav Smirnov, Wendelin Werner
We show how to combine Kesten's scaling relations, the determination of critical exponents associated to the stochastic Loewner evolution process by Lawler, Schramm, and Werner, and Smirnov's proof of Cardy's formula, in order to determine the existence and value of critical exponents associated to percolation on the triangular lattice.
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We prove that critical percolation on any quasi-transitive graph of exponential volume growth does not have a unique infinite cluster. This allows us to deduce from earlier results that critical percolation on any graph in this class does not have any infinite clusters. The result is new when the graph in question is either amenable or nonunimodular.
It has been conjectured in numerous physics papers that in ordinary first-passage percolation on integer lattices, the fluctuation exponent $χ$ and the wandering exponent $ξ$ are related through the universal relation $χ=2ξ-1$, irrespective of the dimension. This is sometimes called the KPZ relation between the two exponents. This article gives a rigorous proof of this conjecture assuming that the exponents exist in a certain sense.
We study first passage percolation on the plane for a family of invariant, ergodic measures on $\mathbb{Z}^2$. We prove that for all of these models the asymptotic shape is the $\ell$-$1$ ball and that there are exactly four infinite geodesics starting at the origin a.s. In addition we determine the exponents for the variance and wandering of finite geodesics. We show that the variance and wandering exponents do not satisfy the relationship of $χ=2ξ-1$ which is expected for independent first passage percolation.
Mixing rate exponent of planar Fortuin-Kasteleyn percolation
该研究定义了共形环系综（CLE）对应的混合速率指数，在FK渗流收敛于CLE的假设下证明二者指数一致，并借助刘维尔量子引力与CLE的耦合推导出CLE$_κ$的混合速率指数公式，解答了前人提出的相关问题。
Consider a long-range percolation model on $\mathbb{Z}^d$ where the probability that an edge $\{x,y\} \in \mathbb{Z}^d \times \mathbb{Z}^d$ is open is proportional to $\|x-y\|_2^{-d-α}$ for some $α>0$ and where $d > 3 \min\{2,α\}$. We prove that in this case the one-arm exponent equals $ \min\{4,α\}/2$. We also prove that the maximal displacement for critical branching random walk scales with the same exponent. This establishes that both models undergo a phase transition in the parameter $α$ when $α=4$.
We examine the percolation model in $\mathbb{Z}^d$ by an approach involving lattice animals, in which their relevant characteristic is surface-area-to-volume ratio. Two critical exponents are introduced. The first is related to the growth rate in size of the number of lattice animals up to translation whose surface-area-to-volume ratio is marginally greater than $1/p_c -1$. The second describes how unusually large clusters form in the percolation model at parameter values slightly below $p_c$. Certain inequalities on the pair of exponents cannot be satisfied, while others imply the continuity …
We prove upper bounds on the one-arm exponent $η_1$ for a class of dependent percolation models which generalise Bernoulli percolation; while our main interest is level set percolation of Gaussian fields, the arguments apply to other models in the Bernoulli percolation universality class, including Poisson-Voronoi and Poisson-Boolean percolation. More precisely, in dimension $d=2$ we prove that $η_1 \le 1/3$ for continuous Gaussian fields with rapid correlation decay (e.g. the Bargmann-Fock field), and in $d \ge 3$ we prove $η_1 \le d/3$ for finite-range fields, both discrete and continuous, a…
We study the probability that the origin is connected to the sphere of radius r (an arm event) in critical percolation in high dimensions, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We prove that this probability decays like 1/r^2. Furthermore, we show that the probability of having k disjoint arms to distance r emanating from the vicinity of the origin is 1/r^2k.
We apply a variation on the methods of Duminil-Copin, Raoufi, and Tassion to establish a new differential inequality applying to both Bernoulli percolation and the Fortuin-Kasteleyn random cluster model. This differential inequality has a similar form to that derived for Bernoulli percolation by Menshikov but with the important difference that it describes the distribution of the volume of a cluster rather than of its radius. We apply this differential inequality to prove the following: The critical exponent inequalities $γ\leq δ-1$ and $Δ\leq γ+1$ hold for percolation and the random cluster m…
All (in)homogeneous bond percolation models on the square, triangular, and hexagonal lattices belong to the same universality class, in the sense that they have identical critical exponents at the critical point (assuming the exponents exist). This is proved using the star-triangle transformation and the box-crossing property. The exponents in question are the one-arm exponent $ρ$, the 2j-alternating-arms exponents $ρ_{2j}$ for $j\ge1$, the volume exponent $δ$, and the connectivity exponent $η$. By earlier results of Kesten, this implies universality also for the near-critical exponents $β$, $…
Backbone exponent for two-dimensional percolation
本文通过SLE气泡测度、刘维尔量子引力与共形场论的可积性，推导出二维临界伯努利渗流骨干指数的精确表达式，揭示其超越数值特性，并推广至所有κ∈(4,8)。
We derive three critical exponents for Bernoulli site percolation on the on the Uniform Infinite Planar Triangulation (UIPT). First we compute explicitly the probability that the root cluster is infinite. As a consequence, we show that the off-critical exponent for site percolation on the UIPT is $β= 1/2$. Then we establish an integral formula for the generating function of the number of vertices in the root cluster. We use this formula to prove that, at criticality, the probability that the root cluster has at least $n$ vertices decays like $n^{-1/7}$. Finally, we also derive an expression fo…
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。
The probability that the cluster of the origin in critical site percolation on the triangular grid has diameter larger than $R$ is proved to decay like $R^{-5/48}$ as $R\to\infty$.
Discrete and continuum Liouville first passage percolation (DLFPP, LFPP) are two approximations of the conjectural $γ$-Liouville quantum gravity (LQG) metric, obtained by exponentiating the discrete Gaussian free field (GFF) and the circle average regularization of the continuum GFF respectively. We show that these two models can be coupled so that with high probability distances in these models agree up to $o(1)$ errors in the exponent, and thus have the same distance exponent. Ding and Gwynne (2018) give a formula for the continuum LFPP distance exponent in terms of the $γ$-LQG dimension exp…
The bulk one-arm exponent for the CLE$_{κ'}$ percolations
研究CLE$_{κ'}$渗流的体单臂指数，通过从BCLE迭代构造CLE渗流及结合刘维尔量子引力与SLE曲线求解该指数，得出临界三态Potts模型双色单臂指数，还得到BCLE相关精确结果及参数关系的新推导。
Discrete Liouville first passage percolation (LFPP) with parameter $ξ> 0$ is the random metric on a sub-graph of $\mathbb Z^2$ obtained by assigning each vertex $z$ a weight of $e^{ξh(z)}$, where $h$ is the discrete Gaussian free field. We show that the distance exponent for discrete LFPP is strictly positive for all $ξ> 0$. More precisely, the discrete LFPP distance between the inner and outer boundaries of a discrete annulus of size $2^n$ is typically at least $2^{αn}$ for an exponent $α> 0$ depending on $ξ$. This is a crucial input in the proof that LFPP admits non-trivial subsequential sca…
Building on the identification of the scaling limit of the critical percolation exploration process as a Schramm-Loewner Evolution, we derive a PDE characterization for the crossing probability of an annulus.
Consider a cellular automaton with state space $\{0,1 \}^{{\mathbb Z}^2}$ where the initial configuration $ω_0$ is chosen according to a Bernoulli product measure, 1's are stable, and 0's become 1's if they are surrounded by at least three neighboring 1's. In this paper we show that the configuration $ω_n$ at time n converges exponentially fast to a final configuration $\barω$, and that the limiting measure corresponding to $\barω$ is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents $β$, $η$, $ν$ and $γ$, and of …
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。
We prove that for Bernoulli percolation on a graph $\mathbb{Z}^2\times\{0,\dots,k\}$ ($k\ge 0$), there is no infinite cluster at criticality, almost surely. The proof extends to finite range Bernoulli percolation models on $\mathbb{Z}^2$ which are invariant under $π/2$-rotation and reflection.
We prove that near-critical percolation and dynamical percolation on the triangular lattice $η\mathbb{T}$ have a scaling limit as the mesh $η\to 0$, in the "quad-crossing" space $\mathcal{H}$ of percolation configurations introduced by Schramm and Smirnov. The proof essentially proceeds by "perturbing" the scaling limit of the critical model, using the pivotal measures studied in our earlier paper. Markovianity and conformal covariance of these new limiting objects are also established.
Parafermionic observables were introduced by Smirnov for planar FK percolation in order to study the critical phase $(p,q)=(p_c(q),q)$. This article gathers several known properties of these observables. Some of these properties are used to prove the divergence of the correlation length when approaching the critical point for FK percolation when $1\le q\le 4$. A crucial step is to consider FK percolation on the universal cover of the punctured plane. We also mention several conjectures on FK percolation with arbitrary cluster-weight $q>0$.
We consider the bond percolation problem on a transient weighted graph induced by the excursion sets of the Gaussian free field on the corresponding cable system. Owing to the continuity of this setup and the strong Markov property of the field on the one hand, and the links with potential theory for the associated diffusion on the other, we rigorously determine the behavior of various key quantities related to the (near-)critical regime for this model. In particular, our results apply in case the base graph is the three-dimensional cubic lattice. They unveil the values of the associated criti…
We give mathematical proofs to a number of statements which appeared in the series of papers by Kleban, Simmons and Ziff where they computed the probabilities of several percolation crossing events.
We prove that in the critical Bernoulli percolation on two dimensional lattice slabs the probabilities of open left-right crossings of rectangles with any given aspect ratio are uniformly positive.
It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE(6). We provide here a detailed proof, which relies on Smirnov's theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy's formula). The version of convergence to SLE(6) that we prove suffices for the Smirnov-Werner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops.
We investigate the maximal non-critical cluster in a big box in various percolation-type models. We investigate its typical size, and the fluctuations around this typical size. The limit law of these fluctuations are related to maxima of independent random variable with law described by a single cluster.
For directed last passage percolation on $\mathbb{Z}^2$ with exponential passage times on the vertices, let $T_{n}$ denote the last passage time from $(0,0)$ to $(n,n)$. We consider asymptotic two point correlation functions of the sequence $T_{n}$. In particular we consider ${\rm Corr}(T_{n}, T_{r})$ for $r\le n$ where $r,n\to \infty$ with $r\ll n$ or $n-r \ll n$. We show that in the former case ${\rm Corr}(T_{n}, T_{r})=Θ((\frac{r}{n})^{1/3})$ whereas in the latter case $1-{\rm Corr}(T_{n}, T_{r})=Θ((\frac{n-r}{n})^{2/3})$. The argument revolves around finer understanding of polymer geometry…
Iterated Graph Systems (II): Bernoulli percolation and critical-exponent universality classes on hierarchical lattices
研究层级晶格上的伯努利渗流，证明有限维数族不能决定普适类，并否证普适性需可公度重整化尺度的猜想。
In high dimensional percolation at parameter $p < p_c$, the one-arm probability $π_p(n)$ is known to decay exponentially on scale $(p_c - p)^{-1/2}$. We show the same statement for the ratio $π_p(n) / π_{p_c}(n)$, establishing a form of a hypothesis of scaling theory. As part of our study, we provide sharp estimates (with matching upper and lower bounds) for several quantities of interest at the critical probability $p_c$. These include the tail behavior of volumes of, and chemical distances within, spanning clusters, along with the scaling of the two-point function at "mesoscopic distance" fr…
Let $G$ be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on $G$. We prove that if $G$ is nonamenable and $p > p_c(G)$ then there exists a positive constant $c_p$ such that \[\mathbf{P}_p(n \leq |K| < \infty) \leq e^{-c_p n}\] for every $n\geq 1$, where $K$ is the cluster of the origin. We deduce the following two corollaries: 1. Every infinite cluster in supercritical percolation on a transitive nonamenable graph has anchored expansion almost surely. This answers positively a question of Benjamini, Lyons, and Schramm (1997). 2. For transitive nonamenabl…
Bootstrap percolation is a wide class of monotone cellular automata with random initial state. In this work we develop tools for studying in full generality one of the three `universality' classes of bootstrap percolation models in two dimensions, termed subcritical. We introduce the new notion of `critical densities' serving the role of `difficulties' for critical models, but adapted to subcritical ones. We characterise the critical probability in terms of these quantities and successfully apply this link to prove new and old results for concrete models such as DTBP and Spiral as well as a ge…
We consider self-avoiding walk and percolation in $\Zd$, oriented percolation in $\Zd\times\Zp$, and the contact process in $\Zd$, with $p D(\cdot)$ being the coupling function whose range is denoted by $L<\infty$. For percolation, for example, each bond $\{x,y\}$ is occupied with probability $p D(y-x)$. The above models are known to exhibit a phase transition when the parameter $p$ varies around a model-dependent critical point $\pc$. We investigate the value of $\pc$ when $d>6$ for percolation and $d>4$ for the other models, and $L\gg1$. We prove in a unified way that $\pc=1+C(D)+O(L^{-2d})$…
We consider a type of dependent percolation introduced by Aizenman and Grimmett, who showed that certain "enhancements" of independent (Bernoulli) percolation, called essential, make the percolation critical probability strictly smaller. In this paper we first prove that, for two-dimensional enhancements with a natural monotonicity property, being essential is also a necessary condition to shift the critical point. We then show that (some) critical exponents and the scaling limit of crossing probabilities of a two-dimensional percolation process are unchanged if the process is subjected to a m…
2D Percolation path exponents $x^{\cal P}_{\ell}$ describe probabilities for traversals of annuli by $\ell$ non-overlapping paths, each on either occupied or vacant clusters, with at least one of each type. We relate the probabilities rigorously to amplitudes of $O(N=1)$ models whose exponents, believed to be exact, yield $x^{\cal P}_{\ell}=({\ell}^2-1)/12$. This extends to half-integers the Saleur--Duplantier exponents for $k=\ell/2$ clusters, yields the exact fractal dimension of the external cluster perimeter, $D_{EP}=2-x^{\cal P}_3=4/3$, and also explains the absence of narrow gate fjords,…
Kesten showed the exponential decay of percolation probability in the subcritical phase for the two-dimensional percolation model. This result implies his celebrated computation that $p_c=0.5$ for bond percolation in the square lattice, and site percolation in the triangular lattice, respectively. In this paper, we present a simpler proof for Kesten's theorem.
We determine the asymptotic speed of the first-passage percolation process on some ladder-like graphs (or width-2 stretches) when the times associated with different edges are independent and exponentially distributed but not necessarily all with the same mean. The method uses a particular Markov chain associated with the first-passage percolation process and properties of its stationary distribution.
This paper has been withdrawn by the author due to a crucial sign error in Proposition 3.1.
We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In particular the expected number of clusters separating two pairs of points converges to an explicit conformal invariant. Our proof is independent of earlier results and $SLE$ techniques, and in principle should provide a new approach to establishing conformal invariance of percolation.
We study scaling limits and conformal invariance of critical site percolation on triangular lattice. We show that some percolation-related quantities are harmonic conformal invariants, and calculate their values in the scaling limit. As a particular case we obtain conformal invariance of the crossing probabilities and Cardy's formula. Then we prove existence, uniqueness, and conformal invariance of the continuum scaling limit.
We review some of the recent progress on the scaling limit of two-dimensional critical percolation; in particular, the convergence of the exploration path to chordal SLE(6) and the "full" scaling limit of cluster interface loops. The results given here on the full scaling limit and its conformal invariance extend those presented previously. For site percolation on the triangular lattice, the results are fully rigorous. We explain some of the main ideas, skipping most technical details.
In this note, we study the model of directed last passage percolation on $\mathbb{Z}^2$, with i.i.d. exponential weight. We consider the maximum paths from vertices $\left(0,\lfloor k^{2/3} \rfloor\right)$ and $(\lfloor k^{2/3} \rfloor,0)$ to $(n,n)$, respectively. For the coalescing point of these paths, we show that the probability for it being $>kR$ far away from the origin is in the order of $R^{-2/3}$. This is motivated by a recent work of Basu, Sarkar, and Sly, where the same estimate was obtained for semi-infinite geodesics, and the optimal exponent for the finite case was left open. Ou…
We investigate the so-called monochromatic arm exponents for critical percolation in two dimensions. These exponents, describing the probability of observing j disjoint macroscopic paths, are shown to exist and to form a different family from the (now well understood) polychromatic exponents. More specifically, our main result is that the monochromatic j-arm exponent is strictly between the polychromatic j-arm and (j+1)-arm exponents.
We study gradient percolation for site percolation on the triangular lattice. This is a percolation model where the percolation probability depends linearly on the location of the site. We prove the results predicted by physicists for this model. More precisely, we describe the fluctuations of the interfaces around their (straight) scaling limits, and the expected and typical lengths of these interfaces. These results build on the recent results for critical percolation on this lattice by Smirnov, Lawler, Schramm and Werner, and on the scaling ideas developed by Kesten.
For two-dimensional percolation at criticality, we discuss the inequality $α_4 > 1$ for the polychromatic four-arm exponent (and stronger versions, the strongest so far being $α_4 \geq 1 + \frac{α_2}{2}$, where $α_2$ denotes the two-arm exponent). We first briefly discuss five proofs (some of them implicit and not self-contained) from the literature. Then we observe that, by combining two of them, one gets a completely self-contained (and yet quite short) proof.
We consider the first-passage percolation problem on effectively one-dimensional graphs with vertex set {1,...,n}\times{0,1} and translation-invariant edge-structure. For three of six non-trivial cases we obtain exact expressions for the asymptotic percolation rate χ by solving certain recursive distributional equations and invoking results from ergodic theory to identify χ as the expected asymptotic one-step growth of the first-passage time from (0,0) to (n,0).
We provide the first nontrivial upper bound for the chemical distance exponent in two-dimensional critical percolation. Specifically, we prove that the expected length of the shortest horizontal crossing path of a box of side length $n$ in critical percolation on $\mathbb{Z}^2$ is bounded by $Cn^{2-δ}π_3(n)$, for some $δ>0$, where $π_3(n)$ is the "three-arm probability to distance $n$." This implies that the ratio of this length to the length of the lowest crossing is bounded by an inverse power of $n$ with high probability. In the case of site percolation on the triangular lattice, we obtain …
Geodesics and Wandering Exponents in Brochette First-Passage Percolation
研究 Brochette 首达渗流模型中测地线的最大横向偏差 H_n，证明测地线的存在性并建立 H_n 的量级，给出依赖环境下明确的徘徊指数。
Let $\{η_{N, v}: v\in V_N\}$ be a discrete Gaussian free field in a two-dimensional box $V_N$ of side length $N$ with Dirichlet boundary conditions. We study the Liouville first passage percolation, i.e., the shortest path metric where each vertex is given a weight of $e^{γη_{N, v}}$ for some $γ>0$. We show that for sufficiently small but fixed $γ>0$, the expected Liouville FPP distance between any pair of vertices is $O(N^{1-γ^2/10^3})$.
We consider planar stationary exponential Last Passage Percolation in the positive quadrant with boundary weights. For $ρ\in (0,1)$ and points $v_N=((1-ρ)^2 N,ρ^2 N)$ going to infinity along the characteristic direction, we establish right tail estimates with the optimal exponent for the exit time of the geodesic, along with optimal exponent estimates for the upper tail moderate deviations for the passage time. For the case $ρ=\frac{1}{2}$ in the stationary model, we establish the lower bound estimate with the optimal exponent for the lower tail of the passage time. Our arguments are based on …
Consider critical site percolation on $\mathbb{Z}^d$ with $d \geq 2$. Cerf (2015) pointed out that from classical work by Aizenman, Kesten and Newman (1987) and Gandolfi, Grimmett and Russo (1988) one can obtain that the two-arms exponent is at least $1/2$. The paper by Cerf slightly improves that lower bound. Except for $d=2$ and for high $d$, no upper bound for this exponent seems to be known in the literature so far (not even implicity). We show that the distance-$n$ two-arms probability is at least $c n^{-(d^2 + 4 d -2)}$ (with $c >0$ a constant which depends on $d$), thus giving an upper …
Strict inequalities for arm exponents in planar percolation
本文提出一种证明相关性不等式定量改进的通用方法，将其应用于方格伯努利键渗流臂估计，证明双臂指数严格大于单臂指数两倍（回答Garban和Steif问题）及单色臂指数严格大于多色版本（非新结果）。
We prove that the critical percolation parameter for Finitary Random Interlacements (FRI) is continuous with respect to the path length parameter $T$. The proof uses a result which is interesting on its own right; equality of natural critical parameters for FRI percolation phase transition.
For the Gaussian free field on a $(d + 1)$-regular tree with $d \geq 2$, we study the percolative properties of its level sets in the critical and the near-critical regime. In particular, we show the continuity of the percolation probability, derive an exact symptotic tail estimate for the cardinality of the connected component of the critical level set, and describe the asymptotic behaviour of the percolation probability in the near-critical regime.
A scaling limit of crossing probabilities for critical percolation on the square lattice
本文证明了正方形晶格上临界渗流在等边三角形中的交叉概率存在标度极限，并指出Cardy公式在该情形下不成立。
In a recent breakthrough work, Chatterjee [Ann. of Math. (2) 177 (2013) 663-697] proved a long standing conjecture that relates the transversal exponent $ξ$ and the fluctuation exponent $χ$ in first-passage percolation on $\mathbb{Z}^d$. The purpose of this paper is to replace the main argument of Chatterjee (2013) and give an alternative proof of this relation. Specifically, we show that under the assumption that exponents defined in Chatterjee (2013) exist, one has the relation $χ\leq2ξ-1$. One advantage of our argument is that it does not require the "nearly Gamma" assumption of Chatterjee …
Let $\{η(v): v\in V_N\}$ be a discrete Gaussian free field in a two-dimensional box $V_N$ of side length $N$ with Dirichlet boundary conditions. We study the Liouville first passage percolation, i.e., the shortest path metric where each vertex is given a weight of $e^{γη(v)}$ for some $γ>0$. We show that for sufficiently small but fixed $γ>0$, with probability tending to $1$ as $N\to \infty$, all geodesics between vertices of macroscopic Euclidean distances simultaneously have (the conjecturally unique) length exponent strictly larger than 1.
We consider a critical Bernoulli site percolation on the uniform infinite planar triangulation. We study the tail distributions of the peeling time, perimeter, and volume of the hull of a critical cluster. The exponents obtained here differs by a factor 2 from those computed previously by Angel and Curien (2015) in the case of critical site percolation on the uniform infinite half-plane triangulation.
Backbone exponent and annulus crossing probability for planar percolation
本文推导出二维逾渗的骨干指数为超越数，并给出两条同色不相交路径穿越环域概率的精确公式，揭示其与共形场论谱及 SLE/LQG 可积结构的联系。
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