Upper tails for triangles
Bobby DeMarco, Jeff Kahn
用$ξ$表示通常的（Erdens-Rényi）随机图$G(m,p)$，$p>1/m$和$η>0$中的三角形个数，我们证明了（对于某些$C_η>0$）$$\Pr(ξ> (1+η)\E ξ) < \exp[-C_η\min{m^2p^2\log(1/p),m^3p^3}].$$这是紧到$C_η$的值.
With $ξ$ the number of triangles in the usual (Erdős-Rényi) random graph $G(m,p)$, $p>1/m$ and $η>0$, we show (for some $C_η>0$) $$\Pr(ξ> (1+η)\E ξ) < \exp[-C_η\min{m^2p^2\log(1/p),m^3p^3}].$$ This is tight up to the value of $C_η$.
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Let $N_{\triangle}(G)$ be the number of triangles in a graph $G$. In [14] and [25] (respectively) the following bounds were proved on the lower tail behaviour of triangle counts in the dense Erdős-Rényi random graphs $G_m\sim G(n,m)$: \[ \mathbb{P}\big(N_{\triangle}(G_m) \, < \, (1-δ)\mathbb{E}[N_{\triangle}(G_m)]\big) \,=\, \exp\left(-Θ\left(δ^2n^3\right)\right) \qquad \text{if $n^{-3/2}\ll δ\ll n^{-1}$} \] and \[ \mathbb{P}\big(N_{\triangle}(G_m) \, < \, (1-δ)\mathbb{E}[N_{\triangle}(G_m)]\big) \,=\, \exp\left(-Θ(δ^{2/3}n^2) \right) \qquad \text{if $n^{-3/4} \ll δ\ll 1$.} \] Neeman, Radin an…
Let $N$ be the number of triangles in an Erdős-Rényi graph $\mathcal{G}(n,p)$ on $n$ vertices with edge density $p=d/n,$ where $d>0$ is a fixed constant. It is well known that $N$ weakly converges to the Poisson distribution with mean ${d^3}/{6}$ as $n\rightarrow \infty$. We address the upper tail problem for $N,$ namely, we investigate how fast $k$ must grow, so that the probability of $\{N\ge k\}$ is not well approximated anymore by the tail of the corresponding Poisson variable. Proving that the tail exhibits a sharp phase transition, we essentially show that the upper tail is governed by P…
Large deviations for triangles in scale-free random graphs
本文研究了无标度非均匀随机图中三角形数量上尾的大偏差估计，发现其概率在 $α<4/3$、$α>4/3$ 和 $α=4/3$ 时分别由多个度数为 $n$ 的顶点、单个特定度数枢纽及有限个枢纽引起，并呈现半指数或多项式衰减的相变现象。
Lower tails for triangles inside the critical window
针对随机图$G(n,p)$不含三角形的概率问题，首次证明$p = c n^{-1/2}$（$c$足够小）时该概率对数的渐近公式，研究三角形下尾大偏差并揭示相变规律，方法结合簇展开等技术。
The missing log in large deviations for triangle counts
通过在指数中引入一个对数因子，解决了Erdos-Renyi随机图中三角形个数上尾的大偏差估计问题.这种证明方法有可能推广到一般的子图数。
What is the probability that the number of triangles in $\mathcal{G}_{n,p}$, the Erdős-Rényi random graph with edge density $p$, is at least twice its mean? Writing it as $\exp[- r(n,p)]$, already the order of the rate function $r(n,p)$ was a longstanding open problem when $p=o(1)$, finally settled in 2012 by Chatterjee and by DeMarco and Kahn, who independently showed that $r(n,p)\asymp n^2p^2 \log (1/p)$ for $p \gtrsim \frac{\log n}n$; the exact asymptotics of $r(n,p)$ remained unknown. The following variational problem can be related to this large deviation question at $p\gtrsim \frac{\log …
Upper exponential inequalities for the tail probabilities of the centered and normalized number of triangles in the Erdös-Rényi graph are obtained, where the probability of every edge is fixed. The result is formulated in terms of random fields.
Random triangle removal
从$n$顶点上的完全图开始，重复删除均匀选择的三角形的边。这个随机过程一旦到达一个无三角形图就终止了，基本问题是估计最终的边数（相当于完成这个过程所需的时间，或者通过随机贪婪算法打包了多少个边不相交的三角形）。Bollobás和Erdás（1990）在对Ramsey数$R(3,t)$的研究的启发下，证明了期望的最终边数具有$n^{3/2}$阶。$o(n^2)$的上界由Spencer（1995）给出，Rödl和Thoma（1996）独立给出。对于变体和推广给出了几个界限（例如，Alon，Kim and Spencer（1997）and Wormald（1999）），而Bollobás和Erdós的原始问题的最著名的上界是由Grable（1997）提出的$n^{7/4+o(1)}$。没有非平凡的下限是可用的。在这里，我们证明了随机三角形删除中的最终边数很可能等于$n^{3/2+o(1)}$，从而证实了Bollobás和Erdés所提出的3/2指数，并与Spencer等人的预测相匹配。对于上界，对于任何固定的$ε>0$，我们通过按规定的方式顺序粘合$O(1/ε)$三角形来构造一族$\exp(O(1/ε))$图，并动态跟踪它们的所有同态，以任意两个顶点为根，直到保留$n^{3/2+ε}$边的点。一个鞅系统建立了这些随机变量的浓度在其类似的平均值在一个随机图与相应的边缘密度，和一个关键的作用…
Building on the techniques from the breakthrough paper of Harel, Mousset and Samotij, which solved the upper tail problem for cliques, we compute the asymptotics of the upper tail for the number of induced copies of the 4-cycle in the binomial random graph $G_{n,p}$. We observe a new phenomenon in the theory of large deviations of subgraph counts. This phenomenon is that, in a certain (large) range of $p$, the upper tail of the induced 4-cycle does not admit a naive mean-field approximation.
Upper tails for counting objects in randomly induced subhypergraphs and rooted random graphs
给出了固定超图H的随机导出子超图的边数的一般上尾估计，并通过矩估计给出了简单的证明。作为应用，我们考虑了整数随机子集上的等差数列和Schur三元组的个数.在本文的第二部分中，我们返回到随机图的子图计数，并在有根的情况下提供上尾估计。
Consider the upper tail probability that the homomorphism count of a fixed graph $H$ within a large sparse random graph $G_n$ exceeds its expected value by a fixed factor $1+δ$. Going beyond the Erdős-Rényi model, we establish here explicit, sharp upper tail decay rates for sparse random $d_n$-regular graphs (provided $H$ has a regular $2$-core), and for sparse uniform random graphs. We further deal with joint upper tail probabilities for homomorphism counts of multiple graphs $H_1,\ldots, H_k$ (extending the known results for $k=1$), and for inhomogeneous graph ensembles (such as the stochast…
Upper tails of subgraph counts in directed random graphs
本文研究有向随机图中有向子图计数的上尾概率，将其转化为边加权有向图上的变分问题，并给出上下界及大量可精确求解的图类。
Suppose that $X$ is a bounded-degree polynomial with nonnegative coefficients on the $p$-biased discrete hypercube. Our main result gives sharp estimates on the logarithmic upper tail probability of $X$ whenever an associated extremal problem satisfies a certain entropic stability property. We apply this result to solve two long-standing open problems in probabilistic combinatorics: the upper tail problem for the number of arithmetic progressions of a fixed length in the $p$-random subset of the integers and the upper tail problem for the number of cliques of a fixed size in the random graph $…
For a sequence of random variables $(X_1, X_2, \ldots, X_n)$, $n \geq 1$, that are independent and identically distributed with a regularly varying tail with index $-α$, $α\geq 0$, we show that the contribution of the maximum term $M_n \triangleq \max(X_1,\ldots,X_n)$ in the sum $S_n \triangleq X_1 + \cdots +X_n$, as $n \to \infty$, decreases monotonically with $α$ in stochastic ordering sense.
For r \ge 2, let X be the number of r-armed stars K_{1,r} in the binomial random graph G_{n,p}. We study the upper tail \Pr(X \ge (1+ε)\E X), and establish exponential bounds which are best possible up to constant factors in the exponent (for the special case of stars K_{1,r} this solves a problem of Janson and Rucinski, and confirms a conjecture by DeMarco and Kahn). In contrast to the widely accepted standard for the upper tail problem, we do not restrict our attention to constant ε, but also allow for ε\ge n^{-α} deviations.
Given a fixed graph H, what is the (exponentially small) probability that the number X_H of copies of H in the binomial random graph G_{n,p} is at least twice its mean? Studied intensively since the mid 1990s, this so-called infamous upper tail problem remains a challenging testbed for concentration inequalities. In 2011 DeMarco and Kahn formulated an intriguing conjecture about the exponential rate of decay of \Pr(X_H \ge (1+ε) \E X_H) for fixed ε>0. We show that this upper tail conjecture is false, by exhibiting an infinite family of graphs violating the conjectured bound.
Upper Tails for Cliques
与$ξ_{k}=ξ_{k}^{n,p}$的$K_k$的副本数在通常的（Erdens-Rényi）随机图$G(n,p)$，$p\geq n^{-2/(k-1)}$和$η>0$，我们表明当$k>1$ $$\Pr(ξ_k> (1+η)\E ξ_k) < \exp [-\gO_{η,k} \min\{n^2p^{k-1}\log(1/p), n^kp^{\binom{k}{2}}\}].$$这是紧的指数常数的值。
On the upper tail of star counts in random graphs
本文研究了随机二项图中 $r$-星图计数的上尾问题，在稀疏条件下确定了其概率渐近行为，首次为不规则图类解决了子图计数上尾问题。
The upper tail problem in a random graph asks to estimate the probability that the number of copies of some fixed subgraph in an Erdős--Rényi random graph exceeds its expectation by some constant factor. There has been much exciting recent progress on this problem. We study the corresponding problem for hypergraphs, for which less is known about the large deviation rate. We present new phenomena in upper tail large deviations for sparse random hypergraphs that are not seen in random graphs. We conjecture a formula for the large deviation rate, i.e., the first order asymptotics of the log-proba…
For a $Δ$-regular connected graph ${\sf H}$ the problem of determining the upper tail large deviation for the number of copies of ${\sf H}$ in $\mathbb{G}(n,p)$, an Erdős-Rényi graph on $n$ vertices with edge probability $p$, has generated significant interests. For $p\ll 1$ and $np^{Δ/2} \gg (\log n)^{1/(v_{\sf H}-2)}$, where $v_{\sf H}$ is the number of vertices in ${\sf H}$, the upper tail large deviation event is believed to occur due to the presence of localized structures. In this regime the large deviation event that the number of copies of ${\sf H}$ in $\mathbb{G}(n,p)$ exceeds its exp…
In the late 1990s, Kim and Vu pioneered an inductive method for showing concentration of certain random variables X. Shortly afterwards, Janson and Ruci{ń}ski developed an alternative inductive approach, which often gives comparable results for the upper tail Pr(X \ge (1+\eps) E[X]). In some cases, both methods yield upper tail estimates which are best possible up to a logarithmic factor in the exponent, but closing this narrow gap has remained a technical challenge. In this paper we present a BK-inequality based combinatorial sparsification idea that can recover this missing logarithmic term …
Universality in prelimiting tail behavior for regular subgraph counts in the Poisson regime
本文证明了在 Poisson 机制下，任意正则子图计数的上尾行为在特定阈值处发生相变，从而将此前仅在三角形情况下已知的相变普适性推广至一般正则图。
Moderate deviations of triangle counts in sparse Erdős-Rényi random graphs $G(n,m)$ and $G(n,p)$
本文研究稀疏 Erdős-Rényi 随机图中三角形计数的中偏差，给出两种模型下上尾对数概率的常数因子估计，并在非局部化区域获得 $G(n,p)$ 的渐近精确结果。
We study the upper tail of the number of arithmetic progressions of a given length in a random subset of {1,...,n}, establishing exponential bounds which are best possible up to constant factors in the exponent. The proof also extends to Schur triples, and, more generally, to the number of edges in random induced subhypergraphs of `almost linear' k-uniform hypergraphs.
Upper tails for homomorphism counts in sparse random hypergraphs
本研究围绕$r$-均匀超图的上尾问题展开，在已有约简为熵变分问题的基础上，证实了Liu和Zhao提出的通用公式猜想对多类超图成立，并证明了满足边覆盖性质的$r$-图计数的通用大偏差上界。
We provide exact large-time equivalents of the density and upper tail distributions of the exponential functional of a subordinator in terms of its Laplace exponents. This improves previous results on the logarithmic asymptotic behaviour of the upper tail. Since the results in their general form remain rather abstract, we study some fairly common situations in details to see how the behavior near 0 of the Lévy measure of the subordinator influences the asymptotics of the density and upper tail distributions. Several examples of applications are then discussed.
An Improved Upper Bound for the Right-Side Tail of the Crossover Distribution at the Edge of the Rarefaction Fan
本文证明了Ivan Corwin和Jeremy Quastel在“稀疏扇边缘的交叉分布”中讨论的分布的右侧尾部的一个改进的上界。
Upper tail bounds for irregular graphs
本文研究稀疏Erdős-Rényi图中不规则子图计数的上尾大偏差，推导了一般不规则图及严格平衡图的偏差率，并针对星图在中间参数范围证明了局部化行为及均值场近似的渐近紧性。
A tail bound for cumulant series for complex functions of independent random variables
本文利用多维差分建立独立随机向量有界复函数累积量级数的显式截断误差界，将既有实函数理论推广至复函数，并给出 Berry–Esseen 界、随机图三角形展开和正则图计数等应用。
For a fixed positive integer $\;k,\;$ limit laws of linearly normalized $\;k$-th upper order statistics are well known. In this article, a comprehensive study of tail behaviours of limit laws of normalized $k$-th upper order statistics under fixed and random sample sizes is carried out using tail equivalence which leads to some interesting tail behaviours of the limit laws. These lead to definitive answers about their max domains of attraction. Stochastic ordering properties of the limit laws are also studied. The results obtained are not dependent on linear norming and apply to power norming …
The areas of Ramsey theory and random graphs have been closely linked ever since Erdős' famous proof in 1947 that the 'diagonal' Ramsey numbers $R(k)$ grow exponentially in $k$. In the early 1990s, the triangle-free process was introduced as a model which might potentially provide good lower bounds for the 'off-diagonal' Ramsey numbers $R(3,k)$. In this model, edges of $K_n$ are introduced one-by-one at random and added to the graph if they do not create a triangle; the resulting final (random) graph is denoted $G_{n,\triangle}$. In 2009, Bohman succeeded in following this process for a positi…
Upper tails for arithmetic progressions revisited
该研究针对前N个正整数的p-偏置随机子集中k项等差数列数量的对数上尾概率给出渐近精确估计，划分出三类参数区域，借助概率论工具和更强版本的算术结构集合熵稳定性组合结果完成推导。
On Upper Bounds for the Tail Distribution of Geometric Sums of Subexponential Random Variables
重新考虑了Kalashnikov和Tsitsiashvili构造次指数和几何和尾分布的上界的方法。通过表达的问题，在一个更概率的光，几个改进和一个校正，这使得构造的界限显着更紧。给出了几个例子，说明如何实现的理论结果。
The KPZ Upper-Tail Field as a KPZ Fixed Point with Brownian Initial Data
本文通过有向景观变分和局部布朗比较，将KPZ上尾场正时间扇区识别为从双边布朗V-型轮廓出发的KPZ不动点，负时间扇区用指数倾斜表示，并恢复其大尺度极限。
An upper tail field of the KPZ fixed point
本文研究了具有narrow-wedge初始条件的KPZ不动点，通过在高点邻域适当放大，构造出定义于全2维空间的上尾场，并证明其在正负时间区间分别表现为KPZ不动点与Brown型场。
We obtain new upper tail probabilities of $m$-times integrated Brownian motions under the uniform norm and the $L^p$ norm. For the uniform norm, Talagrand's approach is used, while for the $L^p$ norm, Zolotare's approach together with suitable metric entropy and the associated small ball probabilities are used. This proposed method leads to an interesting and concrete connection between small ball probabilities and upper tail probabilities (large ball probabilities) for general Gaussian random variable in Banach spaces. As applications, explicit bounds are given for the largest eigenvalue of t…
Asymptotic normality for triangle counting in the sparse $β$-model
本文研究了稀疏β模型中三角形数量的渐近正态性，通过异质参数向量的范数确定其渐近均值和方差，并利用马尔蒂尼-斯坦方法推导了与标准正态分布的Kolmogorov距离上界，最终证明了其渐近正态性。
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.
Suppose that $(Z_n)_{n\geq0}$ is a supercritical branching process in independent and identically distributed random environment. The right tail function of the scaled growth rate for $(Z_n)_{n\geq0}$ is studied. The upper bounds for $\displaystyle\mathbb{P}\left[\frac{\log Z_n}{Mn}-μ\geq x\right]$ for any $x\geq3$ are obtained, by applying an extension of the Hoeffding type inequalities.
Triangle-Saturated Graphs in the Semi-Random Graph Process
本文针对半随机图过程的在线与离线版本，研究构造三角饱和图所需轮次的上下界，明确了该图性质下算法的轮次需求范围。
This paper investigates an upper bound of the operator norm for sub-Gaussian tailed random matrices. A lot of attention has been put on uniformly bounded sub-Gaussian tailed random matrices with independent coefficients. However, little has been done for sub-Gaussian tailed random matrices whose matrix coefficients variance are not equal or for matrix for which coefficients are not independent. This is precisely the subject of this paper. After proving that random matrices with uniform sub-Gaussian tailed independent coefficients satisfy the Tracy Widom bound, that is, their matrix operator no…
In this paper we consider the limiting distribution of KPZ growth models with random but not stationary initial conditions introduced in [Chhita-Ferrari-Spohn 2018]. The one-point distribution of the limit is given in terms of a variational problem. By directly studying it, we deduce the right tail asymptotic of the distribution function. This gives a rigorous proof and extends the results obtained in [Meerson-Schmidt 2017].
We present a proof of an upper tail bound of the correct order (up to a constant factor in the exponent) in two classes of stationary models in the KPZ universality class. The proof is based on an exponential identity due to Rains in the case of Last Passage Percolation with exponential weights, and recently re-derived by Emrah-Jianjigian-Seppäiläinen (EJS). Our proof follows very similar lines for the two classes of models we consider, using only general monotonocity and convexity properties, and can thus be expected to apply to many other stationary models.
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.
Comparison for upper tail probabilities of random series
设$\{ξ_n\}$是独立同分布随机变量序列。本文研究了两个上尾概率$\mathbb{P}\{\sum_{n=1}^{\infty}a_n|ξ_n|^p\geq r\}$和$\mathbb{P}\{\sum_{n=1}^{\infty}b_n|ξ_n|^p\geq r\}$作为$r\rightarrow\infty$与两个不同的实序列$\{a_n\}$和$\{b_n\}.$的比较。第一个结果是高斯随机变量$\{ξ_n\},$的，在这种情况下，这两个概率是等价的适当的缩放。第二个结果是对更一般的随机变量，从而证明了等价的一种较弱形式（即对数水平）。
Value-at-Risk, Tail Value-at-Risk and upper tail transform of the sum of two counter-monotonic random variables
本文研究两个任意反单调随机变量之和的风险度量分解问题，推广已有结果，给出了VaR、TVaR和止损变换的分解公式。
Upper large deviations for Branching Processes in Random Environment with heavy tails
随机环境中的分支过程（BPRE）$(Z_n:n\geq0)$是Galton Watson过程的推广，其中在每一代中，再生律在i.i.d.中随机选取。方式当再生律可能有重尾时，我们在这里确定过程的上大偏差。BPRE的行为与环境的相关随机游动有关，其增量分布类似于后代分布的对数平均值。我们得到了$(Z_n:n\geq0)$的上速率函数的表达式，即当$-\log \mathbb{P}(Z_n\geq e^{θn})/n$。它取决于环境的相关随机游动的速率函数、对数生存成本$γ:=-\lim_{n\to\infty} \log \mathbb{P}(Z_n>0)/n$和$Z_1$尾部分布的多项式衰减$β$。我们给出解释这个速率函数的过程中，达到非常大的值$(Z_n: n \geq 0)$的代价最小的方式，并描述了相变。然后，我们推导出再生律没有重尾时的速率函数，这推广了Böinghoff和Kersting（2009）以及Bansaye和Berestycki（2008）关于上大偏差的结果。最后，我们给出了Galton Watson过程重尾的上大偏差。
We derive upper bounds on the tail conditional expectation of binomial and Poisson random variables. Those upper bounds are subsequently employed to the problem of obtaining non-asymptotic lower bounds on the probability that the aforementioned random variables are significantly larger than their expectation.
Annealed upper tails for the energy of a polymer
研究了随机带电对称瞬态随机游动能量的上尾。我们假设只有在同一位置上的电荷两两相互作用。我们考虑退火估计，也就是说，当我们平均两个随机性，在三维或更多。我们得到了一个大偏差原理和一个显式的速率函数的一大类电荷分布。
Upper tail large deviations of the directed landscape
从单点尾部界出发，在度量层面建立有向景观的上尾大偏差原理，速率函数由测度的 Kruzhkov 熵给出，并应用于证明有向测地线的大偏差原理。
Consider a random vector $U$, whose distribution function coincides in its upper tail with that of an Archimedean copula. We report the fact that the conditional distribution of $U$, conditional on one of its components, has under a mild condition on the generator function independent upper tails, no matter what the unconditional tail behavior is. This finding is extended to Archimax copulas.
Upper tails of self-intersection local times of random walks: survey of proof techniques
$\Z^d$上随机游动的自相交局部时大大超过其期望值的概率的渐近性是一个令人着迷的课题，因为它与从统计力学到大偏差理论和变分分析的一些模型有关，并且因为可以观察到的效应的多样性。然而，证明的上界是出了名的困难，需要各种复杂的技术。我们调查了一些化学和一些最近制定的技术和结果。这是12月6日至10日在Luminy举行的CIRM会议上关于{ \it多余的自相交本地时间和相关主题}的演讲的扩展摘要，2010.
We consider the asymmetric simple exclusion process (ASEP) on $\mathbb{Z}$ started from step initial data and obtain the exact Lyapunov exponents for $H_0(t)$, the integrated current of ASEP. As a corollary, we derive an explicit formula for the upper-tail large deviation rate function for $-H_0(t)$. Our result matches with the rate function for the integrated current of the totally asymmetric simple exclusion process (TASEP) obtained in [Johansson 00](arXiv:math/9903134).
An upper bound of the lower tail of the mass of balls under the critical $2d$ stochastic heat flow
本文研究了二维临界随机热流的球体质量下尾上界，证明了其对数的可积性及严格正性，回答了关于局部行为的开放问题。
General upper and lower tail estimates using Malliavin calculus and Stein's equations
根据Ivan Nourdin和Giovanni Peccati最近开发的策略，我们提供了一种通用技术来比较给定随机变量与参考分布的尾部。这使我们能够给出具体的条件，以确保各种幂或指数类型的随机变量的尾部上界和/或下界。Nourdin-Peccati策略分析了Stein方法和Malliavin演算之间的关系，并适用于处理与高斯定律的比较。通过详细研究一般Stein方程的解的行为，我们表明，该策略可以扩展到比较广泛的一类法律，包括许多皮尔逊分布。
Upper tail large deviations for Brownian motions with one-sided collisions
本文针对属于KPZ普适类的单侧反射相互作用布朗运动系统，推导了三种基本初始条件下的上尾大偏差原理并给出速率函数显式形式，发现周期情形下速率函数含Fredholm行列式公式中已有的Lambert-W函数。
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…
A New Type Of Upper And Lower Bounds On Right-Tail Probabilities Of Continuous Random Variables
本文提出了一种新型上下界，用于界定具有无界和左侧半有界支撑的连续随机变量的右尾概率，该边界仅依赖PDF、一阶导数及两个参数，并在数值例子中验证了其紧致性。
We consider the upper and lower tail probabilities for the centered (by time$/24$) and scaled (according to KPZ time$^{1/3}$ scaling) one-point distribution of the Cole-Hopf solution of the KPZ equation when started with initial data drawn from a very general class. For the lower tail, we prove an upper bound which demonstrates a crossover from super-exponential decay with exponent $3$ in the shallow tail to an exponent $5/2$ in the deep tail. For the upper tail, we prove super-exponential decay bounds with exponent $3/2$ at all depth in the tail.
In this paper we consider the problem of estimating the joint upper and lower tail large deviations of the edge eigenvalues of an Erdős-Rényi random graph $\mathcal{G}_{n,p}$, in the regime of $p$ where the edge of the spectrum is no longer governed by global observables, such as the number of edges, but rather by localized statistics, such as high degree vertices. Going beyond the recent developments in mean-field approximations of related problems, this paper provides a comprehensive treatment of the large deviations of the spectral edge in this entire regime, which notably includes the well…
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