We consider solutions to the maximum recursion on weighted branching trees given by$$X\,{\buildrel d\over=}\,\bigvee_{i=1}^{N}{A_iX_i}\vee B,$$where $N$ is a random natural number, $B$ and $\{A_i\}_{i\in\mathbb{N}}$ are random positive numbers and $X_i$ are independent copies of $X$, also independent of $N$, $B$, $\{A_i\}_{i\in\mathbb{N}}$. Properties of solutions to this equation are governed mainly by the function $m(s)=\mathbb{E}\big[\sum_{i=1}^NA_i^s\big]$. Recently, Jelenković and Olvera-Cravioto proved, assuming e.g. $m(s)<1$ for some $s$, that the asymptotic behavior of the endogenous solution $R$ to the above equation is power-law, i.e.$$\mathbb{P}[R>t]\sim Ct^{-α}$$for some $α>0$ and $C>0$. In this paper we assume $m(s)\ge 1$ for all $s$ and prove analogous results.
Tail exponents of the three-dimensional uniform spanning tree and Abelian sandpile
Beijing International Center for Mathematical Research, Peking University（北京大学数学中心）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Graduate School of Informatics, Kyoto University（京都大学情报学府）
A Markov tree is a random vector indexed by the nodes of a tree whose distribution is determined by the distributions of pairs of neighbouring variables and a list of conditional independence relations. Upon an assumption on the tails of the Markov kernels associated to these pairs, the conditional distribution of the self-normalized random vector when the variable at the root of the tree tends to infinity converges weakly to a random vector of coupled random walks called tail tree. If, in addition, the conditioning variable has a regularly varying tail, the Markov tree satisfies a form of one…
Tail bounds for the height and width of a random tree with a given degree sequence
固定序列c=（c_1，...，c_n）的和为n-1的非负整数。我们说一棵有根树T有子序列c，如果T的节点可以排序为v_1，.，v_n，使得对于每个1 <= i <= n，v_i恰好有c_i个孩子。设T是从所有具有子序列c的平面树中随机均匀抽取的平面树。在本文中，我们证明了T的高度（任何节点的最大深度）和宽度（任何单个深度处的最大节点数）的亚高斯尾界。当c满足c_1^2+.+时，这些界限是最优的，直到指数为常数。c_n^2=O（n）;后者可以被看作是子序列的“有限方差”条件。
We find the exponential exact two-terms non-asymptotic expression for the maximum and minimum distribution of a non-Gaussian, in general case, random vector.
The tail of the maximum of Brownian motion minus a parabola
分析了布朗运动的最大值N减去一条抛物线的尾部行为，并给出了P（N>x）在x趋于无穷大时的渐近展开.这扩展了Huesler和Piterbarg（1999）关于尾部行为的一阶结果。我们还指出了Groeneboom（2010）和Janson，Louchard and Martin-Löf（2010）中某些结果之间的关系。
This article investigates the heat kernel of the two-dimensional uniform spanning tree. We improve previous work by demonstrating the occurrence of log-logarithmic fluctuations around the leading order polynomial behaviour for the on-diagonal part of the quenched heat kernel. In addition we give two-sided estimates for the averaged heat kernel, and we show that the exponents that appear in the off-diagonal parts of the quenched and averaged versions of the heat kernel differ. Finally, we derive various scaling limits for the heat kernel, the implications of which include enabling us to sharpen…
A weighted recursive tree is an evolving tree in which vertices are assigned random vertex-weights and new vertices connect to a predecessor with a probability proportional to its weight. Here, we study the maximum degree and near-maximum degrees in weighted recursive trees when the vertex-weights are almost surely bounded and their distribution function satisfies a mild regularity condition near zero. We are able to specify higher-order corrections to the first order growth of the maximum degree established in prior work. The accuracy of the results depends on the behaviour of the weight dist…
We provide the exact large-time behavior of the tail distribution of the extinction time of a self-similar fragmentation process with a negative index of self-similarity, improving thus a previous result on the logarithmic asymptotic behavior of this tail. Two factors influence this behavior: the distribution of the largest fragment at the time of a dislocation and the index of self-similarity. As an application we obtain the asymptotic behavior of all moments of the largest fragment and compare it to the behavior of the moments of a tagged fragment, whose decrease is in general significantly …
The discrete membrane model is a Gaussian random interface whose inverse covariance is given by the discrete biharmonic operator on a graph. In literature almost all works have considered the field as indexed over $\mathbb{Z}^d$, and this enabled one to study the model using methods from partial differential equations. In this article we would like to investigate the dependence of the membrane model on a different geometry, namely trees. The covariance is expressed via a random walk representation which was first determined by Vanderbei (1984). We exploit this representation on $m$-regular tre…
This is the first of a three paper series in which we present a comprehensive study of the extreme value theory of the scale-inhomogeneous discrete Gaussian free field. This model was introduced by Arguin and Ouimet who computed the first order of the maximum. In this first paper we establish tail estimates for the maximum value, which allow to deduce the log-correction to the order of the maximum and tightness of the centred maximum. Our proofs are based on the second moment method and Gaussian comparison techniques.
We study the one-dimensional branching random walk in the case when the step size distribution has a stretched exponential tail, and, in particular, no finite exponential moments. The tail of the step size $X$ decays as $\mathbb{P}[X \geq t] \sim a \exp(-λt^r)$ for some constants $a, λ> 0$ where $r \in (0,1)$. We give a detailed description of the asymptotic behaviour of the position of the rightmost particle, proving almost-sure limit theorems, convergence in law and some integral tests. The limit theorems reveal interesting differences betweens the two regimes $ r \in (0, 2/3)$ and $ r \in (…
We prove large deviation results for the position of the rightmost particle, denoted by $M_n$, in a one-dimensional branching random walk in a case when Cramér's condition is not satisfied. More precisely we consider step size distributions with stretched exponential upper and lower tails, i.e.~both tails decay as $e^{-|t|^r}$ for some $r\in( 0,1)$. It is known that in this case, $M_n$ grows as $n^{1/r}$ and in particular faster than linearly in $n$. Our main result is a large deviation principle for the laws of $n^{-1/r}M_n$ . In the proof we use a comparison with the maximum of (a random num…
Extreme local statistics in random graphs: maximum tree extension counts
Czech Technical University in Prague（布拉格捷克理工大学）；Institute of Computer Science of the Czech Academy of Sciences（捷克科学院计算机研究所）；Vilnius University（维尔纽斯大学）；University of California San Diego（加州大学圣地亚哥分校）
