Transmission and navigation on disordered lattice networks, directed spanning forests and Brownian web

Critical random forests

Let $F(N,m)$ denote a random forest on a set of $N$ vertices, chosen uniformly from all forests with $m$ edges. Let $F(N,p)$ denote the forest obtained by conditioning the Erdos-Renyi graph $G(N,p)$ to be acyclic. We describe scaling limits for the largest components of $F(N,p)$ and $F(N,m)$, in the critical window $p=N^{-1}+O(N^{-4/3})$ or $m=N/2+O(N^{2/3})$. Aldous described a scaling limit for the largest components of $G(N,p)$ within the critical window in terms of the excursion lengths of a reflected Brownian motion with time-dependent drift. Our scaling limit for critical random forests …

The Gorin-Shkolnikov identity and its random tree generalization

In a recent pair of papers Gorin and Shkolnikov (2018) and Hariya (2016) have shown that the area under normalized Brownian excursion minus one half the integral of the square of its total local time is a centered normal random variable with variance $\frac{1}{12}$. Gaudreau Lamarre and Shkolnikov (2019) generalized this to Brownian bridges, and ask for a combinatorial interpretation. We provide a combinatorial interpretation using random forests on $n$ vertices. In particular, we show that there is a process level generalization for a certain infinite forest model. We also show analogous resu…

Encoding multitype Galton-Watson forests and a multitype Ray-Knight theorem

We provide a simple forest model to encode the genealogical structure of a multitype Galton-Watson process with immigration. We provide two encodings of these forests by stochastic processes. We show, under appropriate conditions, the depth-first encodings of each particular type converge to a solution to a system of stochastic integral equations involving height processes perturbed by functionals of their local times. The forest picture allows us to extend the Ray-Knight theorem and show that local time of the solution to the system of equations form a multitype continuous state branching pro…

The Brownian disk viewed from a boundary point

We provide a new construction of Brownian disks in terms of forests of continuous random trees equipped with nonnegative labels corresponding to distances from a distinguished point uniformly distributed on the boundary of the disk. This construction shows in particular that distances from the distinguished point evolve along the boundary as a five-dimensional Bessel bridge. As an important ingredient of our proofs, we show that the uniform measure on the boundary, as defined in the earlier work of Bettinelli and Miermont, is the limit of the suitably normalized volume measure on a small tubul…

Scaling limit of random forests with prescribed degree sequences

In this paper, we consider the random plane forest uniformly drawn from all possible plane forests with a given degree sequence. Under suitable conditions on the degree sequences, we consider the limit of a sequence of such forests with the number of vertices tends to infinity in terms of Gromov-Hausdorff-Prokhorov topology. This work falls into the general framework of showing convergence of random combinatorial structures to certain Gromov-Hausdorff scaling limits, described in terms of the Brownian Continuum Random Tree, pioneered by the work of Aldous. In fact we identify the limiting rand…

莱维森林的雷骑士定理的随机积分表示

Stochastic integral representations for the Ray-Knight theorem of the Levy forest

本文提出了一种新的随机积分表示，用于描述莱维森林的雷骑士定理，并扩展了最近关于布朗运动局部时间漂移的研究。

Collision times of random walks and applications to the Brownian web

Convergence of directed forests, spanning on random subsets of lattices or on point processes, towards the Brownian web has made the subject of an abundant literature, a large part of which relies on a criterion proposed by Fontes, Isopi, Newman and Ravishankar (2004). One of their convergence condition, called (B2), states that the probability of the event that there exists three distinct paths for a time interval of length $t(>0)$, all starting within a segment of length $\varepsilon$, is of small order of $\varepsilon$. This condition is often verified by applying an FKG type correlation in…

随机扩展式决策 I：随机决策森林

Decision making in stochastic extensive form I: Stochastic decision forests

本文提出随机决策森林理论，用单次抽签替代“自然”博弈者，以桥接决策树与滤流概念，并构造时间索引行动路径类，为扩展式随机微分博弈的逼近理论奠基。

随机生成树的部分采样

Sampling biased spanning forests through marked vertices

研究完全图K_N的κ偏 spanning forest 中连接多个顶点的子树分布及其渐进行为，并在κ=O(√N)处识别相变现象。

随机标记树中垂直边轮廓的马尔可夫性质

Markov properties for the vertical edge profile in random labelled trees

研究随机标记树中垂直边轮廓的马尔可夫性质，揭示其与超布朗运动的关联。

Scaling limit of a drainage network model on perturbed lattice

Study of random networks generally requires the nodes to be independently and uniformly distributed such as a Poisson point process. In this work, we venture beyond this standard paradigm and investigate a stochastic forest obtained from a drainage network model constructed on a randomly perturbed subset of $\mathbb{Z}^2$, where both horizontal and vertical perturbations are given by exponentially decaying unbounded discrete random variables and vertical perturbations are allowed in the upward direction only. We show that the resultant stochastic network is a single tree a.s. We further establ…

有向生成森林：合并与维度的关系

The Directed Spanning Forest: coalescence versus dimension

本研究探讨了任意维度和不同 $p$ 取值下 $\ell^p$ 有向生成森林的不相交树数量，证明了其在三维及二维特定条件下的树结构，并引入随机支配论证将结果推广至非平面和奇异情形。

非均匀平面排水网络中的吞吐量

Throughput in inhomogeneous planar drainage networks

本文研究非均匀平面排水网络中导航方案的吞吐量，证明总流量与树长联合收敛于时间非均匀布朗运动下的面积，直至首次触零，对应极限最大路径长度。

The scaling limit of the minimum spanning tree of the complete graph

Consider the minimum spanning tree (MST) of the complete graph with n vertices, when edges are assigned independent random weights. Endow this tree with the graph distance renormalized by n^{1/3} and with the uniform measure on its vertices. We show that the resulting space converges in distribution, as n tends to infinity, to a random measured metric space in the Gromov-Hausdorff-Prokhorov topology. We additionally show that the limit is a random binary R-tree and has Minkowski dimension 3 almost surely. In particular, its law is mutually singular with that of the Brownian continuum random tr…