Concentration of Broadcast Models on Trees

Tree/Endofunction Bijections and Concentration Inequalities

We demonstrate a method for proving precise concentration inequalities in uniformly random trees on $n$ vertices, where $n\geq1$ is a fixed positive integer. The method uses a bijection between mappings $f\colon\{1,\ldots,n\}\to\{1,\ldots,n\}$ and doubly rooted trees on $n$ vertices. The main application is a concentration inequality for the number of vertices connected to an independent set in a uniformly random tree, which is then used to prove partial unimodality of its independent set sequence. So, we give probabilistic arguments for inequalities that often use combinatorial arguments.

使用随机矩阵的广播

Broadcasting with Random Matrices

受自旋玻璃理论启发，研究树和稀疏随机图上的重构问题，通过引入新估计量并利用Markov链工具解决多层随机性挑战，确定了广播模型的重构阈值。

Concentration of Lipschitz functionals of determinantal and other strong Rayleigh measures

Let X_1 ,..., X_n be a collection of binary valued random variables and let f : {0,1}^n -> R be a Lipschitz function. Under a negative dependence hypothesis known as the {\em strong Rayleigh} condition, we show that f - E f satisfies a concentration inequality generalizing the classical Gaussian concentration inequality for sums of independent Bernoullis: P (S_n - E S_n > a) < exp (-2 a^2 / n). The class of strong Rayleigh measures includes determinantal measures, weighted uniform matroids and exclusion measures; some familiar examples from these classes are generalized negative binomials and …

A concentration inequality for the maximum vertex degree in random dissections

We obtain a concentration inequality for the maximum degree of a vertex in a uniformly random dissection of a polygon. This resolves a conjecture posed by Curien and Kortchemski in 2012. Our approach is based on a bijection with dual trees and the tools of analytic combinatorics.

Phylogenetic mixtures: Concentration of measure in the large-tree limit

The reconstruction of phylogenies from DNA or protein sequences is a major task of computational evolutionary biology. Common phenomena, notably variations in mutation rates across genomes and incongruences between gene lineage histories, often make it necessary to model molecular data as originating from a mixture of phylogenies. Such mixed models play an increasingly important role in practice. Using concentration of measure techniques, we show that mixtures of large trees are typically identifiable. We also derive sequence-length requirements for high-probability reconstruction.

Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees

We study the effect of boundary conditions on the relaxation time of the Glauber dynamics for the hard-core model on the tree. The hard-core model is defined on the set of independent sets weighted by a parameter $λ$, called the activity. The Glauber dynamics is the Markov chain that updates a randomly chosen vertex in each step. On the infinite tree with branching factor $b$, the hard-core model can be equivalently defined as a broadcasting process with a parameter $ω$ which is the positive solution to $λ=ω(1+ω)^b$, and vertices are occupied with probability $ω/(1+ω)$ when their parent is uno…

随机块模型的精确相变与树上的重构

Exact Phase Transitions for Stochastic Block Models and Reconstruction on Trees

该研究通过树与图过程的新耦合方法及树上广播过程的精细分析，严格论证了随机块模型中q=3、q=4社区情形的Kesten-Stigum界相关猜想，明确了不同社区数量下的计算-统计差距与界的紧性特征。

Canonical Trees, Compact Prefix-free Codes and Sums of Unit Fractions: A Probabilistic Analysis

For fixed $t\ge 2$, we consider the class of representations of $1$ as sum of unit fractions whose denominators are powers of $t$ or equivalently the class of canonical compact $t$-ary Huffman codes or equivalently rooted $t$-ary plane "canonical" trees. We study the probabilistic behaviour of the height (limit distribution is shown to be normal), the number of distinct summands (normal distribution), the path length (normal distribution), the width (main term of the expectation and concentration property) and the number of leaves at maximum distance from the root (discrete distribution).