We define an analog of Plancherel measure for the set of rooted unlabeled trees on n vertices, and a Markov chain which has this measure as its stationary distribution. Using the combinatorics of commutation relations, we show that order n^2 steps are necessary and suffice for convergence to the stationary distribution.
We give a general construction of triangulations starting from a walk in the quarter plane with small steps, which is a discrete version of the mating of trees. We use a special instance of this construction to give a bijection between maps equipped with a rooted spanning tree and walks in the quarter plane. We also show how the construction allows to recover several known bijections between such objects in a uniform way.
The Perron value $ρ(T)$ of a rooted tree $T$ has a central role in the study of the algebraic connectivity and characteristic set, and it can be considered a weight of spectral nature for $T$. A different, combinatorial weight notion for $T$ - the moment $μ(T)$ - emerges from the analysis of Kemeny's constant in the context of random walks on graphs. In the present work, we compare these two weight concepts showing that $μ(T)$ is "almost" an upper bound for $ρ(T)$ and the ratio $μ(T)/ρ(T)$ is unbounded but at most linear in the order of $T$. To achieve these primary goals, we introduce two new…
A sharp density bound for 5-connected graphs with no $\Ke$ minor
Guangdong University of Petrochemical Technology（广东石油化工学院）；Lingnan Normal University（岭南师范学院）；Fujian Institute of Education（福建教育学院）；China University of Geosciences（中国地质大学）
We consider random partitions of the vertex set of a given finite graph that can be sampled by means of loop-erased random walks stopped at a random exponential time of parameter $q>0$. The related random blocks tend to cluster nodes visited by the random walk on time scale $1/q$. This random partitioning is induced by a measure of rooted spanning forest of the graph, which generalizes the classical uniform spanning tree measure and which can be obtained as a zero-limit of FK-percolation with an external cemetery state. Some general properties of this rooted forest measure and related determin…
A rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph $\mathcal{G}(n,r,d)$. In particular, we find the sharp threshold for balanced binary trees. More generally, we show that all sequences of balanced trees with uniformly bounded degrees and height tending to infinity appear above a sharp threshold, and none of these appears below the same value. Our results hold more generally for geometric graphs satisfying a mild…
Cutoff Phenomenon for Random Walks on Kneser Graphs
遍历马尔可夫链的截断现象描述了在一个可忽略的时间段内（称为截断窗口）收敛到其平稳分布的急剧转变。研究了Kneser图上简单随机游动的截断现象，Kneser图是一类遍历马氏链。给定两个整数$n$和$k$，Kneser图$K(2n+k,n)$定义为顶点集为$n$大小的$\{1,\ldots,2n+k\}$的所有子集，且如果$A\cap B =\emptyset$，则两个顶点$A$和$B$由一条边连接的图.我们表明，对于任何$k=O(n)$，$K(2n+k,n)$上的随机游走表现出一个截止在$\frac{1}{2}\log_{1+k/n}{(2n+k)}$与大小为$O(\frac{n}{k})$的窗口。
