S. Friedli, B. N. B. de Lima, V. Sidoravicius
Consider independent long range percolation on $\mathbf{Z}^2$, where horizontal and vertical edges of length $n$ are open with probability $p_n$. We show that if $\limsup_{n\to\infty}p_n>0,$ then there exists an integer $N$ such that $P_N(0\leftrightarrow \infty)>0$, where $P_N$ is the truncated measure obtained by taking $p_{N,n}=p_n$ for $n \leq N$ and $p_{N,n}=0$ for all $n> N$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we obtain optimal uniform lower tail estimates for the probability distribution of the properly scaled length of the longest up/right path of the last passage site percolation model considered by Johansson in [12]. The estimates are used to prove a lower tail moderate deviation result for the model. The estimates also imply the convergence of moments, and also provide a verification of the universal scaling law relating the longitudinal and the transversal fluctuations of the model.
We answer a question of Ahlberg and Steif (2014) by finding the tail behaviour of the crossing probability in near-critical planar percolation. Interestingly, this superexponentially small behaviour is different from the case of dynamical percolation, where the analogous tail probability was proved to be at least exponential and at most superpolynomial by Hammond, Mossel and Pete (2012). The proof is simple, given the scale covariance established by Garban, Pete and Schramm (2013).
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…
Consider first passage percolation with identical and independent weight distributions and first passage time ${\rm T}$. In this paper, we study the upper tail large deviations $\mathbb{P}({\rm T}(0,nx)>n(μ+ξ))$, for $ξ>0$ and $x\neq 0$ with a time constant $μ$ and a dimension $d$, for weights that satisfy a tail assumption $ β_1\exp{(-αt^r)}\leq \mathbb P(τ_e>t)\leq β_2\exp{(-αt^r)}.$ When $r\leq 1$ (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as $\exp{(-(2dξ+o(1))n)}$. When $1< r\leq d$, we find that the rate function can be naturally d…
One-point fluctuations for exponential last passage percolation under upper-tail conditioning
该研究针对上尾条件下的指数型定向最后通行渗流，确定了特定空间区域内及边界的单点涨落规律，扩展了相关工作，并通过串联队列对应关系揭示了非典型离站对网络其他位置的影响。
For first passage percolation on $\mathbb{Z}^2$ with i.i.d. bounded edge weights, we consider the upper tail large deviation event; i.e., the rare situation where the first passage time between two points at distance $n$, is macroscopically larger than typical. It was shown by Kesten (1986) that the probability of this event decays as $\exp (-Θ(n^2))$. However the question of existence of the rate function i.e., whether the log-probability normalized by $n^2$ tends to a limit, had remained open. We show that under some additional mild regularity assumption on the passage time distribution, the…
We show that oriented percolation occurs whenever a condition is satisfied called "exponential intersection tails". This condition says that a measure on paths exists for which the probability of two independent paths intersecting in more than k sites is exponentially small in k.
We prove that the heavy clusters are indistinguishable for Bernoulli percolation on quasi-transitive nonunimodular graphs. As an application, we show that the uniqueness threshold of any quasi-transitive graph is also the threshold for connectivity decay. This resolves a question of Lyons and Schramm (1999) in the Bernoulli percolation case and confirms a conjecture of Schonmann (2001). We also prove that every infinite cluster of Bernoulli percolation on a nonamenable quasi-transitive graph is transient almost surely.
Distribution tail of the distance to infinity in supercritical bond percolation
该研究针对Z^d上首次通过渗流，在权重分布有界且0处有超临界原子的条件下，估计了0到无穷远通行时间的分布尾部，还证明了点到点通行时间上尾偏差的速率函数在0附近渐近为多项式，其特殊情形涉及超临界键渗流的不交割集数量。
This paper presents a new, short proof of the computation of the upper tail large deviation rate function for the Brownian directed percolation model. Through a distributional equivalence between the last passage time in this model and the largest eigenvalue in a random matrix drawn from the Gaussian Unitary Ensemble, this provides a new proof a previously known result. The method leads to associated results for the stationary Brownian directed percolation model which have not been observed before.
We consider last-passage percolation models in two dimensions, in which the underlying weight distribution has a heavy tail of index alpha<2. We prove scaling laws and asymptotic distributions, both for the passage times and for the shape of optimal paths; these are expressed in terms of a family (indexed by alpha) of "continuous last-passage percolation" models in the unit square. In the extreme case alpha=0 (corresponding to a distribution with slowly varying tail) the asymptotic distribution of the optimal path can be represented by a random self-similar measure on [0,1], whose multifractal…
Consider a Markov process ω_t at equilibrium and some event C (a subset of the state-space of the process). A natural measure of correlations in the process is the pairwise correlation \Pr[ω_0,ω_t \in C] - \Pr[ω_0 \in C]^2. A second natural measure is the probability of the continual occurrence event \{ω_s \in C, \forall s\in [0,t]\}. We show that for reversible Markov chains, and any event C, pairwise decorrelation of the event C implies a decay of the probability of the continual occurrence event \{ω_s \in C, \forall s \in [0,t]\} as t\to\infty. We provide examples showing that our results a…
The areas under workload process and under queuing process in a single server queue over the busy period have many applications not only in queuing theory but also in risk theory or percolation theory. We focus here on the tail behaviour of distribution of these two integrals. We present various open problems and conjectures, which are supported by partial results for some special cases.
In this paper we consider the first passage percolation with identical and independent exponentially distributions, called the Eden growth model, and we study the upper tail large deviations for the first passage time ${\rm T}$. Our main results prove that for any $ξ>0$ and $x\neq 0$, $\mathbb{P}({\rm T}(0,nx)>n(μ+ξ))$ decays as $\exp{(-(2dξ+o(1))n)}$ with a time constant $μ$ and a dimension $d$. Moreover, we extend the result to stretched exponential distributions. On the contrary, we construct a continuous distribution with a finite exponential moment where the rate function does not exist.
Optimal tail exponents in general last passage percolation via bootstrapping & geodesic geometry
本文针对一般权重分布的二维最后通过渗流，利用超可加性、测度集中及测地线西瓜几何，通过迭代自助法证明测地线权重具有 $3/2$ 与 $3$ 的最优上下尾指数。
We consider oriented long-range percolation on a graph with vertex set $\mathbb{Z}^d \times \mathbb{Z}_+$ and directed edges of the form $\langle (x,t), (x+y,t+1)\rangle$, for $x,y$ in $\mathbb{Z}^d$ and $t \in \mathbb{Z}_+$. Any edge of this form is open with probability $p_y$, independently for all edges. Under the assumption that the values $p_y$ do not vanish at infinity, we show that there is percolation even if all edges of length more than $k$ are deleted, for $k$ large enough. We also state the analogous result for a long-range contact process on $\mathbb{Z}^d$.
We study moderate deviations in the exponential corner growth model, both in the bulk setting and the increment-stationary setting. The main results are sharp right-tail bounds on the last-passage time and the exit point of the increment-stationary process. The arguments utilize calculations with the stationary version and a moment generating function identity due to E. Rains, for which we give a short probabilistic proof. As applications of the deviation bounds, we derive upper bounds on the speed of distributional convergence in the Busemann function and competition interface limits.
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 …
Inhomogeneous percolation on the hierarchical configuration model with a heavy-tailed degree distribution
本文研究了重尾度分布层次配置模型上的非均匀渗流，利用稀疏 Lévy 过程描述连通分量大小的极限，并为带质量和权重的乘法凝聚建立了 Feller 型性质。
We prove several facts concerning Lipschitz percolation, including the following. The critical probability p_L for the existence of an open Lipschitz surface in site percolation on Z^d with d\ge 2 satisfies the improved bound p_L \le 1-1/[8(d-1)]. Whenever p > p_L, the height of the lowest Lipschitz surface above the origin has an exponentially decaying tail. The lowest surface is dominated stochastically by the boundary of a union of certain independent, identically distributed random subsets of Z^d. As a consequence, for p sufficiently close to 1, the connected regions of Z^{d-1} above which…
Consider critical site percolation on a "nice" planar lattice: each vertex is occupied with probability $p = p_c$, and vacant with probability $1 - p_c$. Now, suppose that additional vacancies ("holes", or "impurities") are created, independently, with some small probability, i.e. the parameter $p_c$ is replaced by $p_c - \varepsilon$, for some small $\varepsilon > 0$. A celebrated result by Kesten says, informally speaking, that on scales below the characteristic length $L(p_c - \varepsilon)$, the connection probabilities remain of the same order as before. We prove a substantial and subtle g…
On the upper tail large deviation rate function for chemical distance in supercritical percolation
本文研究了超临界键渗流中化学距离的上尾大偏差问题，证明了当 $d\ge 3$ 且 $\varepsilon>0$ 足够小时速率函数的存在性，并揭示该大偏差事件由时空切点迫使测地线绕行引发。
Quenched scaling limit of critical percolation clusters on Galton-Watson trees
研究临界渗流簇在超临界Galton-Watson树上的淬火标度极限，证明其与α-稳定树一致，同时获得簇大小尾部的淬火渐进行为。
Branching random walks on multidimensional lattice with heavy tails and a constant branching rate are considered. It is shown that under these conditions (heavy tails and constant rate), the front propagates exponentially fast, but the particles inside of the front are distributed very non-uniformly. The particles exhibit intermittent behavior in a large part of the region behind the front (i.e., the particles are concentrated only in very sparse spots there). The zone of non-intermittency (were particles are distributed relatively uniformly) extends with a power rate. This rate is found.
We study large Wigner random matrices in the case when the marginal distributions of matrix entries have heavy tails. We prove that the largest eigenvalues of such matrices have Poisson statistics.
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.
We consider the first passage percolation model on the ${\bf Z}^d$ lattice. In this model, we assign independently to each edge $e$ a non-negative passage time $t(e)$ with a common distribution $F$. Let $a_{0,n}$ be the passage time from the origin to $(n,0,..., 0)$. Under the exponential tail assumption, Kesten (1993) and Talagrand (1995) investigated the concentration of $a_{0,n}$ from its mean using different methods. With this concentration and the exponential tail assumption, Alexander gave an estimate for the convergence rate for ${\bf E} a_{0,n}$. In this paper, focusing on a moment con…
Heavy repulsion of clusters in Bernoulli percolation
研究（非）幺模拟可迁图上的 Bernoulli$(p)$ 渗流，证明任意两个重团簇间距离为 1 的顶点集为轻集，推广了 Timár 定理并运用测度类保持等价关系等新工具克服了非幺模设定的挑战。
In the present article we consider a natural generalization of Hammersley's Last Passage Percolation (LPP) called Entropy-controlled Last Passage Percolation (E-LPP), where points can be collected by paths with a global (entropy) constraint which takes in account the whole structure of the path, instead of a local ($1$-Lipschitz) constraint as in Hammersley's LPP. The E-LPP turns out to be a key ingredient in the context of the directed polymer model when the environment is heavy-tailed, which we consider in the related paper [Berger and Torri, 2018]. We prove several estimates on the E-LPP in…
We address the important question of the extent to which random variables and vectors with truncated power tails retain the characteristic features of random variables and vectors with power tails. We define two truncation regimes, soft truncation regime and hard truncation regime, and show that, in the soft truncation regime, truncated power tails behave, in important respects, as if no truncation took place. On the other hand, in the hard truncation regime much of "heavy tailedness" is lost. We show how to estimate consistently the tail exponent when the tails are truncated, and suggest stat…
Finiteness of the percolation threshold for inhomogeneous long-range models in one dimension
该研究针对一维非均匀长程空间随机图，引入系数$\delta_\text{eff}$量化重尾度对长程连接的影响，证明了渗流阈值存在的临界条件，补充了已有相关研究结果，并为有限图巨分支存在性提供判定准则。
We study tail behaviour of the distribution of the area under the positive excursion of a random walk which has negative drift and heavy-tailed increments. We determine the asymptotics for tail probabilities for the area.
Heavy Loads and Heavy Tails
本文利用变换方法研究重尾队列的平稳工作量，揭示了无限方差下尾部渐近性与重流量极限的联系，并证明减载等价性质可推广至重流量极限。
We construct an example of a continuous centered random process with light tails of finite-dimensional distribution but with (relatively) heavy tail of maximum distribution. The apparatus for tails comparison are embedding results for Orlicz and Grand Lebesgue Spaces (GLS).
We study heavy-tailed Hermitian random matrices that are unitarily invariant. The invariance implies that the eigenvalue and eigenvector statistics are decoupled. The motivating question has been whether a freely stable random matrix has stable eigenvalue statistics for the largest eigenvalues in the tail. We investigate this question through the use of both numerical and analytical means, the latter of which makes use of the supersymmetry method. A surprising behaviour is uncovered in that a freely stable random matrix does not necessarily yield stable statistics and if it does then it might …
Large deviations of the giant in supercritical kernel-based spatial random graphs
本文研究超临界空间随机图中巨分量大小的大偏差，证明长程边可将下尾多项式速度指数从(d-1)/d提高到任意ζ_*∈((d-1)/d,1)，并刻画了上尾在幂律度分布下的对数速度及度齐次模型中的线性速度。
Large deviation principle at speed $n$ for the random metric in first-passage percolation
本文研究 $\mathbb Z^d$ 标准首达渗流中盒内重标度随机度量的下尾大偏差，证明其在指数矩条件下满足速度 $n$ 的大偏差原理，并给出速率函数的三种表达及弱矩条件下的概率估计。
First-Passage Percolation on Spread-out line graphs: Microscopic Regime
该研究针对ℓ固定、n→∞的ℓ-扩展线图首通渗流，通过枢轴节点分解方法，得出通行时间Tₙ的大数定律、高斯CLT或非高斯稳定律收敛结果，还证明了最小路径边数的相关极限定理，揭示了Tₙ波动的分布相变。
Only Segmented Heavy Tails Can Produce a Light-Tailed Minimum
本文研究了重尾随机变量的最小值为何能成为轻尾分布的条件，证明了只有特定的重尾分布才能生成轻尾的最小值，并扩展了相关结论。
The purpose of this paper is to show that the use of heavy-tailed distributions in Financial problems is theoretically baseless and can lead to significant misunderstandings. The reason for this the authors see in an incorrect interpretation of the concept of the distributional tail. In accordance with this, in applications it is necessary to use instead of the distributional tail the "smearing" of its central part. keywords: heavy tailed distributions, exponential tails, pre-limit theorems, Financial indexes
We extend the construction principle of phase-type (PH) distributions to allow for inhomogeneous transition rates and show that this naturally leads to direct probabilistic descriptions of certain transformations of PH distributions. In particular, the resulting matrix distributions enable to carry over fitting properties of PH distributions to distributions with heavy tails, providing a general modelling framework for heavy-tail phenomena. We also illustrate the versatility and parsimony of the proposed approach for the modelling of a real-world heavy-tailed fire insurance dataset.
Optimal tail estimates in $β$-ensembles and applications to last passage percolation
本文为 $\beta$-系综最大特征值建立带匹配常数的双边中等偏差估计，并应用于平面指数末段渗流，得到三个迭代对数定律，解决 Ledoux 猜想。
This work is denoted to studying the tail behavior of Cox-Ingersoll-Ross (CIR) processes with regime-switching. One essential difference shown in this work between CIR process with regime-switching and without regime-switching is that the stationary distribution for CIR process with regime-switching could be heavy-tailed. Our results provide a theoretical evidence of the existence of regime-switching for interest rates model based on its heavy-tailed empirical evidence. In this work, we first provide sharp criteria to justify the existence of stationary distribution for the CIR process with re…
We obtain an uniform tail estimates for natural normed sums of independent random variables (r.v.) with regular varying tails of distributions. We give also many examples on order to show the exactness of offered estimates and discuss some applications in the method Monte-Carlo and statistics, and obtain the sufficient conditions for Central and stable limit theorem in the Banach space of continuous function. There are considered a slight generalization on a random variables with super-heavy tails and martingale difference scheme.
In this paper, we will give a sufficient condition for a non-negative random variable $X$ to be heavy tailed by investigating the Laplace-Stieltjes transform of the probability distribution function. We focus on the relation between the singularity at the real point of the axis of convergence and the asymptotic decay of the tail probability. Our theorem is a kind of Tauberian theorems.
Recently, Kigami's resistance form framework has been applied to provide a general approach for deriving the scaling limits of random walks on graphs with a fractal scaling limit. As an illustrative example, this article describes an application to the random conductance model with heavy tails on nested fractal graphs.
As well known, for a supercritical Galton-Watson process $Z_n$ whose offspring distribution has mean $m>1$, the ratio $W_n:=Z_n/m^n$ has a.s. limit, say $W$. We study tail behaviour of the distributions of $W_n$ and $W$ in the case where $Z_1$ has heavy-tailed distribution, that is, $\E e^{λZ_1}=\infty$ for every $λ>0$. We show how different types of distributions of $Z_1$ lead to different asymptotic behaviour of the tail of $W_n$ and $W$. We describe the most likely way how large values of the process occur.
Two of the most popular approximations for the distribution of the steady-state waiting time, $W_{\infty}$, of the M/G/1 queue are the so-called heavy-traffic approximation and heavy-tailed asymptotic, respectively. If the traffic intensity, $ρ$, is close to 1 and the processing times have finite variance, the heavy-traffic approximation states that the distribution of $W_{\infty}$ is roughly exponential at scale $O((1-ρ)^{-1})$, while the heavy tailed asymptotic describes power law decay in the tail of the distribution of $W_{\infty}$ for a fixed traffic intensity. In this paper, we assume a …
Distributions with a heavy tail are difficult to estimate. If the design of a scheduling policy is sensitive to the details of heavy tail distributions of the service times, an approximately optimal solution is difficult to obtain. This paper shows that the optimal scheduling of an M/G/1 queue with heavy tailed service times does not present this difficulty and that an approximately optimal strategy can be derived by truncating the distributions.
DynaSPLIT: Strong tail-optimal scheduling for the M/G/n with heavy-tailed job sizes
本文针对作业规模为重尾分布的M/G/n队列，提出首个适用于高负载的强尾最优调度策略DynaSPLIT，填补了该场景高负载下强尾最优调度的研究空白，证明该队列全稳定区域均可实现强尾最优性。
We introduce and study a model of percolation with constant freezing (PCF) where edges open at constant rate 1, and clusters freeze at rate αindependently of their size. Our main result is that the infinite volume process can be constructed on any amenable vertex transitive graph. This is in sharp contrast to models of percolation with freezing previously introduced, where the limit is known not to exist. Our interest is in the study of the percolative properties of the final configuration as a function of α. We also obtain more precise results in the case of trees. Surprisingly the algebraic …
We prove a functional central limit theorem for partial sums of symmetric stationary long range dependent heavy tailed infinitely divisible processes with a certain type of negative dependence. Previously only positive dependence could be treated. The negative dependence involves cancellations of the Gaussian second order. This leads to new types of limiting processes involving stable random measures, due to heavy tails, Mittag-Leffler processes, due to long memory, and Brownian motions, due to the Gaussian second order cancellations.
We investigate a way of comparing and classifying tails of random variables. Our approach extends the notion of classical indices, such as exponential and moment indices, which are widely used measuring heaviness of tail functions. A non-parametric risk measure applicable for all heavy-tailed random variables is obtained as a concave function that represents the decay speed of tail function. Many key properties of the distribution of a random variable are encoded into this function, which enables a new way to estimate tails. The latter half of the paper is devoted to numerous examples illustra…
We study a generalization of site percolation on a simple cubic lattice, where not only single sites are removed randomly, but also entire parallel columns of sites. We show that typical clusters near the percolation transition are very anisotropic, with different scaling exponents for the sizes parallel and perpendicular to the columns. Below the critical point there is a Griffiths phase where cluster size distributions and spanning probabilities in the direction parallel to the columns have power law tails with continuously varying non-universal powers. This region is very similar to the Gri…
Any random variable with right-unbounded distributional support is the minimum of independent and very heavy-tailed random variables
本文证明任意具有右无界支撑的轻尾随机变量均可表示为两个独立重尾随机变量的最小值，且该结论可推广至有限个独立变量及相依情形。
This paper studies the tail probability of weighted sums of the form $\sum_{i=1}^n c_i X_i$, where random variables $X_i$'s are either independent or pairwise quasi-asymptotical independent with heavy tails. Using $h$-insensitive function, the uniform asymptotic equivalence of the tail probabilities of $\sum_{i=1}^n c_iX_i$, $\max_{1\le k\le n}\sum_{i=1}^k c_iX_i$ and $\sum_{i=1}^n c_iX_i^+$ is established, where $X_i$'s are independent and follow the long-tailed distribution, and $c_i$'s take value in a broad interval. Some further uniform asymptotic results for the weighted sums of $X_i$'s w…
Let $\{X_1, X_2, ... \}$ be a sequence of dependent heavy-tailed random variables with distributions $F_1, F_2,...$ on $(-\infty,\infty)$, and let $τ$ be a nonnegative integer-valued random variable independent of the sequence $\{X_k, k \ge 1\}$. In this framework, we study the asymptotic behavior of the tail probabilities of the quantities $X_{(n)} = \max_{1\le k \le n} X_k$, $S_n =\sum_{k=1}^n X_k$ and $S_{(n)}=\max_{1\le k\le n} S_k$ for $n>1$, and for those of their randomized versions $X_{(τ)}$, $S_τ$ and $S_{(τ)}$. We also consider applications of the results obtained to some commonly-us…
Computation of extreme quantiles and tail-based risk measures using standard Monte Carlo simulation can be inefficient. A method to speed up computations is provided by importance sampling. We show that importance sampling algorithms, designed for efficient tail probability estimation, can significantly improve Monte Carlo estimators of tail-based risk measures. In the heavy-tailed setting, when the random variable of interest has a regularly varying distribution, we provide sufficient conditions for the asymptotic relative error of importance sampling estimators of risk measures, such as Valu…
Asymptotic behavior of the extremal position in a multi-type branching random walk with heavy-tailed displacements
本文研究了位移具有重尾的多类型分支随机游动，探讨了不可约与可约情形下最右粒子的渐近行为，并为不同设定确定了正确的归一化。
We establish some asymptotic expansions for infinite weighted convolution of distributions having regular varying tails. Various applications to statistics and probability are developed.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。