Width and mode of the profile for some random trees of logarithmic height

A central limit theorem for square ice

We prove that the height function associated with the uniform six-vertex model (or equivalently, the uniform homomorphism height function from $\mathbb Z^2$ to $\mathbb Z$) satisfies a central limit theorem, upon some logarithmic rescaling.

Universal height and width bounds for random trees

We prove non-asymptotic stretched exponential tail bounds on the height of a randomly sampled node in a random combinatorial tree, which we use to prove bounds on the heights and widths of random trees from a variety of models. Our results allow us to prove a conjecture and settle an open problem of Janson (https://doi.org/10.1214/11-PS188), and nearly prove another conjecture and settle another open problem from the same work (up to a polylogarithmic factor). The key tool for our work is an equivalence in law between the degrees along the path to a random node in a random tree with given degr…

Sub-exponential tail bounds for conditioned stable Bienaymé-Galton-Watson trees

We establish uniform sub-exponential tail bounds for the width, height and maximal outdegree of critical Bienaymé-Galton-Watson trees conditioned on having a large fixed size, whose offspring distribution belongs to the domain of attraction of a stable law. This extends results obtained for the height and width by Addario-Berry, Devroye & Janson in the finite variance case.

Most trees are short and fat

This work proves new probability bounds relating to the height, width, and size of Galton-Watson trees. For example, if $T$ is any Galton-Watson tree, and $H$, $W$, and $|T|$ are the height, width, and size of $T$, respectively, then $H/W$ has sub-exponential tails and $H/|T|^{1/2}$ has sub-Gaussian tails. Although our methods apply without any assumptions on the offspring distribution, when information is provided about the distribution the method can be adapted accordingly, and always seems to yield tight bounds.

Heavy tails and one-dimensional localization

We address the fundamental questions concerning the operator \begin{eqnarray*} H^{θ_0}ψ(x)=-ψ"(x)+V(x,ω)ψ(x),\,ψ(0)\cosθ_0-ψ'(0)\sinθ_0=0. \end{eqnarray*} where the random potential $V$ has a variety of forms. In one example, it is composed of width one bumps of random heights where the square root of the heights are in the domain of attraction of a stable law with index $α\in(0,1)$ or in another it is composed of width one bumps of height one where the distance between bumps is in the domain of attraction of a stable law with index $α\in(0,1).$ We consider the existence of Lyapunov exponents,…

从倾斜边界条件下六顶点高度函数的对数离域到 Ashkin-Teller、广义随机簇及 $(q_\sigma,q_\tau)$-cubic 模型的减弱穿越概率估计

From logarithmic delocalization of the six-vertex height function under sloped boundary conditions to weakened crossing probability estimates for the Ashkin-Teller, generalized random-cluster, and $(q_σ,q_τ)$-cubic models

本文研究倾斜边界条件下六顶点模型高度函数的 RSW 穿越估计，分析倾斜 Gibbs 态及大长宽比矩形中的冻结困难，以支撑更广泛边界条件下的对数离域证明。

Plancherel随机树的渐近高度

Asymptotic height of Plancherel random trees

研究Plancherel随机树的高度渐进行为，证明其以对数速率增长，并通过变分原理确定相关常数。

临界树既不太短也不太胖

Critical trees are neither too short nor too fat

本文研究了临界尺寸条件Bienaymé树，建立了高度的下尾界和宽度的上尾界，证明了其最优性，并得到了Cauchy分布吸引域内后代分布的精确渐近性，同时提出了相关渐近行为的问题。

Freezing transition of the directed polymer in a $1+d$ random medium : location of the critical temperature and unusual critical properties

In dimension $d \geq 3$, the directed polymer in a random medium undergoes a phase transition between a free phase and a disorder dominated phase. For the latter, Fisher and Huse have proposed a droplet theory based on the scaling of the free energy fluctuations $ΔF(l) \sim l^θ$. On the other hand, in related growth models belonging to the KPZ universality class, Forrest and Tang have found that the height-height correlation function is logarithmic at the transition. For the directed polymer model at criticality, this translates into logarithmic free energy fluctuations $ΔF_{T_c}(l) \sim (\ln …