arXiv subjects

Meltem Ünel

Publications and source records attributed to Meltem Ünel.

7 recordsLinked to original sources

arXiv2025

On exponentially height-penalized random trees

Given $n \in \mathbb{N}$ and $μ\in \mathbb{R}$, a $\textit{$μ$-height-biased tree of size $n$}$ is a random plane tree $\mathbf{\mathbf{T}}_n$ with $n$ vertices with law given by $\mathbb{P}(\mathbf{T}=t) \propto e^{-μh(t)}$, where $t$ ranges over fixed plane trees with $n$ vertices, and $h(t)$ is the height of $t$. Fix a sequence $(μ_n)_{n \ge 1}$ of real numbers, and for $n \ge 1$ let $\mathbf{T}_n$ be a $μ$-height-biased tree of size $n$. Durhuus and Ünel (2023) described the asymptotic behaviour of $h(\mathbf{T}_n)$ when $μ_n \equiv μ\in \mathbb{R}$ is fixed. In this work, we extend their results to arbitrary sequences of positive parameters depending on $n$. Most notably, we show that such a tree behaves like a height-biased Continuum Random Tree (CRT) when $μ_n$ is of order $1/\sqrt{n}$; that its height is asymptotically $(2π^2n/μ_n)^{1/3}$ when $μ_n$ is of larger order than $1/\sqrt{n}$ and of smaller order than $n$; and that its height converges to a fixed constant when $μ_n$ is of order at least $n$, with some random jumps under specific conditions on $μ_n$. We additionally prove various results on second order behaviours, and large deviation principles for the height, for different regimes of $μ_n$. Finally, we describe new statistics of these trees, covering their widths, their root degrees, and the local structure around their roots.

math.PR↗

arXiv2025

Ideal Poisson-Voronoi tessellations on hyperbolic spaces

We study the limit in low intensity of Poisson--Voronoi tessellations in hyperbolic spaces $ \mathbb{H}_{d}$ for $d \geq 2$. In contrast to the Euclidean setting, a limiting nontrivial ideal tessellation $ \mathcal{V}_{d}$ appears as the intensity tends to $0$. The tessellation $ \mathcal{V}_{d}$ is a natural, isometry-invariant decomposition of $ \mathbb{H}_{d}$ into countably many unbounded polytopes, each with a unique end. We study its basic properties, in particular, the geometric features of its cells.

math.PR↗

arXiv2023

Relativistic Lee Model and its Resolvent Analysis

We reexamine the relativistic 2+1 dimensional Lee model in light-front coordinates on flat space and on a space-time with a spatial section given by a compact manifold in the usual canonical formalism. The simpler 2+1 dimension is chosen because renormalization is needed only for the mass difference but not required for the coupling constant and the wavefunction. The model is constructed non-perturbatively based on the resolvent formulation [1]. The bound state spectrum is studied through its ``principal operator" and bounds for the ground state energy are obtained. We show that the formal expression found indeed defines the resolvent of a self-adjoint operator--the Hamiltonian of the interacting system. Moreover, we prove an essential result that the principal operator corresponds to a self-adjoint holomorphic family of type-A in the sense of Kato.

math-ph↗

arXiv2022

Local limits of one-sided trees

A finite \emph{one-sided tree} of height $h$ is defined as a rooted planar tree obtained by grafting branches on one side, say the right, of a spine, i.e. a linear path of length $h$ starting at the root, such that the resulting tree has no simple path starting at the root of length greater than $h$. We consider the distribution $τ_N$ on the set of one-sided trees $T$ of fixed size $N$, such that the weight of $T$ is proportional to $e^{-μh(T)}$, where $μ$ is a real constant and $h(T)$ denotes the height of $T$. We show that, for $N$ large, $τ_N$ has a weak limit as a probability measure supported on infinite one-sided trees. The dependence of the limit measure $τ$ on $μ$ shows a transition at $μ_0=-\ln 2$ from a single spine phase for $μ\leq μ_0$ to a multi-spine phase for $μ> μ_0$. Correspondingly, there is a transition in the volume growth rate of balls around the root as a function of radius from linear growth for $μ<μ_0$, to quadratic growth at $μ=μ_0$, and to qubic growth for $μ> μ_0$.

math.PR↗

arXiv2022

Trees with exponential height dependent weight

We consider planar rooted random trees whose distribution is even for fixed height $h$ and size $N$ and whose height dependence is of exponential form $e^{-μh}$. Defining the total weight for such trees of fixed size to be $Z^{(μ)}_N$, we determine its asymptotic behaviour for large $N$, for arbitrary real values of $μ$. Based on this we evaluate the local limit of the corresponding probability measures and find a transition at $μ=0$ from a single spine phase to a multi-spine phase. Correspondingly, there is a transition in the volume growth rate of balls around the root as a function of radius from linear growth for $μ<0$ to the familiar quadratic growth at $μ=0$ and to cubic growth for $μ> 0$.

math.PR↗

arXiv2022

Trees with power-like height dependent weight

We consider planar rooted random trees whose distribution is even for fixed height $h$ and size $N$ and whose height dependence is given by a power function $h^α$. Defining the total weight for such trees of fixed size to be $Z_N$, a detailed analysis of the analyticity properties of the corresponding generating function is provided. Based on this, we determine the asymptotic form of $Z_N$ and show that the local limit at large size is identical to the Uniform Infinite Planar Tree, independent of the exponent $α$ of the height distribution function.

math.PR↗

arXiv2021

Critical behaviour of loop models on causal triangulations

We introduce a dense and a dilute loop model on causal dynamical triangulations. Both models are characterised by a geometric coupling constant $g$ and a loop parameter $α$ in such a way that the purely geometric causal triangulation model is recovered for $α=1$. We show that the dense loop model can be mapped to a solvable planar tree model, whose partition function we compute explicitly and use to determine the critical behaviour of the loop model. The dilute loop model can likewise be mapped to a planar tree model; however, a closed-form expression for the corresponding partition function is not obtainable using the standard methods employed in the dense case. Instead, we derive bounds on the critical coupling $g_c$ and apply transfer matrix techniques to examine the critical behaviour for $α$ small.

hep-th↗