Bobby DeMarco, Jeff Kahn
With $ξ_{k}=ξ_{k}^{n,p}$ the number of copies of $K_k$ in the usual (Erdős-Rényi) random graph $G(n,p)$, $p\geq n^{-2/(k-1)}$ and $η>0$, we show when $k>1$ $$\Pr(ξ_k> (1+η)\E ξ_k) < \exp [-\gO_{η,k} \min\{n^2p^{k-1}\log(1/p), n^kp^{\binom{k}{2}}\}].$$ This is tight up to the value of the constant in the exponent.
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We study the size of the largest clique $ω(G(n,α))$ in a random graph $G(n,α)$ on $n$ vertices which has power-law degree distribution with exponent $α$. We show that for `flat' degree sequences with $α>2$ whp the largest clique in $G(n,α)$ is of a constant size, while for the heavy tail distribution, when $0<α<2$, $ω(G(n,α))$ grows as a power of $n$. Moreover, we show that a natural simple algorithm whp finds in $G(n,α)$ a large clique of size $(1+o(1))ω(G(n,α))$ in polynomial time.
Clique packings in random graphs
本文研究了稠密 Erdős-Rényi 随机图中边不相交近最大团的数量，通过随机图过程和微分方程法证明了其最大团填充规模的上下界均为 $Θ(n^2/(\log{n})^3)$。
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.
Limit theorems under heavy-tailed scenario in the age dependent random connection models
本文研究年龄依赖随机连接模型中的子图计数极限定理，证明子树与团计数在相应条件下弱收敛到稳定随机变量，并给出团计数期望渐近。
General upper tail estimates are given for counting edges in a random induced subhypergraph of a fixed hypergraph H, with an easy proof by estimating the moments. As an application we consider the numbers of arithmetic progressions and Schur triples in random subsets of integers. In the second part of the paper we return to the subgraph counts in random graphs and provide upper tail estimates in the rooted case.
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…
Asymptotics for cliques in scale-free random graphs
本文研究Chung-Lu非齐次随机图模型中团数的渐近性，针对权重尾部概率为$h^{1-α}l(h)$（$α>2$）的情形，给出了所有团大小及非整数$α$的完整渐近结果，并解释了整数$α$的差异及部分结果。
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 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.
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 $C_η$.
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.
In this paper, we relate the problem of finding a maximum clique to the intersection number of the input graph (i.e. the minimum number of cliques needed to edge cover the graph). In particular, we consider the maximum clique problem for graphs with small intersection number and random intersection graphs (a model in which each one of $m$ labels is chosen independently with probability $p$ by each one of $n$ vertices, and there are edges between any vertices with overlaps in the labels chosen). We first present a simple algorithm which, on input $G$ finds a maximum clique in $O(2^{2^m + O(m)} …
On the clique number of random Cayley graphs and related topics
本文用纯组合方法证明随机凯莱图的团数高概率为 O(log N log log N)，改进了 Alon 的上界，并据此证明 Alon 关于凯莱图团数与独立数均为 O(log N) 的猜想对几乎所有阿贝尔群成立，还构造了自补凯莱图以强形式解决 Alon–Orlitsky 问题。
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 …
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.
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.
Finding cliques in random graphs and the closely related "planted" clique variant, where a clique of size t is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best known polynomial-time algorithms only solve the problem for t = Theta(sqrt(n)). Here we show that beating sqrt(n) would require substantially new algorithmic ideas, by proving a lower bound for the problem in the sum-of-squares (or Lasserre) hierarchy, the most powerful class of semi-definite programming algorithms we know of: r rounds of the sum-of-squares h…
This note gives a proof of an improved upper bound for the right-side tail of a distribution discussed in "Crossover Distributions at the Edge of the Rarefaction Fan" by Ivan Corwin and Jeremy Quastel.
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 approach used by Kalashnikov and Tsitsiashvili for constructing upper bounds for the tail distribution of a geometric sum with subexponential summands is reconsidered. By expressing the problem in a more probabilistic light, several improvements and one correction are made, which enables the constructed bound to be significantly tighter. Several examples are given, showing how to implement the theoretical result.
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。
Peripheral Traps and Lower Bounds on Mixing Times for Random Walks on Sparse Heavy-Tailed Random Intersection Graphs
本文针对稀疏重尾随机交图上的随机游走，分析其混合时间下界，发现外围陷阱导致混合时间下界为Ω(log²n)，局部总变差距离非集中衰减，进而指出不存在全局截断。
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…
Normal approximation for subgraph counts in age-dependent random connection models
本文针对年龄依赖随机连接模型，在轻尾条件下证明团计数的多变量定量正态逼近，并利用相伴随机变量中心极限定理得到子树计数的渐近正态性。
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.
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…
Large deviation behavior of the largest eigenvalue $λ_1$ of Gaussian networks (Erdős-Rényi random graphs $\mathcal{G}_{n,p}$ with i.i.d. Gaussian weights on the edges) has been the topic of considerable interest. Recently in [6,30], a powerful approach was introduced based on tilting measures by suitable spherical integrals, particularly establishing a non-universal large deviation behavior for fixed $p<1$ compared to the standard Gaussian ($p=1$) case. The case when $p\to 0$ was however completely left open with one expecting the dense behavior to hold only until the average degree is logarit…
We study the upper tails for the energy of a randomly charged symmetric and transient random walk. We assume that only charges on the same site interact pairwise. We consider annealed estimates, that is when we average over both randomness, in dimension three or more. We obtain a large deviation principle, and an explicit rate function for a large class of charge distributions.
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。
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).
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。
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。
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…
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