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Christina Goldschmidt

Publications and source records attributed to Christina Goldschmidt.

At least 19 recordsLinked to original sources
arXiv2026

The stable trees revisited

We introduce a new, relatively simple, line-breaking construction of the $α$-stable tree which realises its random finite-dimensional distributions. This is a direct analogue of Aldous' line-breaking construction of the Brownian continuum random tree, which is based on an inhomogeneous Poisson process. Here, we replace the deterministic rate function from the Brownian setting by a random rate process, given by a certain measure-changed $(α-1)$-stable subordinator. Rather than attaching uniformly, the line-segments now connect to locations chosen with probability proportional to the sizes of the jumps of the rate process. We also give a new proof of an invariance principle originally due to Duquesne, which states that the family tree of a Bienaymé branching process with critical offspring distribution in the domain of attraction of an $α$-stable law (for $α\in (1,2))$, conditioned to have $n$ vertices, converges on rescaling distances appropriately to the $α$-stable tree. Our proof makes use of a discrete line-breaking construction of the branching process tree, which we show converges to our continuous line-breaking construction.

math.PR↗
arXiv2025

Discrete snakes with globally centered displacements

We prove a scaling limit for globally centered discrete snakes on size-conditioned critical Bienaymé trees. More specifically, under a global finite variance condition, we prove convergence in the sense of random finite-dimensional distributions of the head of the discrete snake (suitably rescaled) to the head of the Brownian snake driven by a Brownian excursion. When the third moment of the offspring distribution is finite, we further prove uniform functional convergence under a necessary tail condition on the displacements. We also consider displacement distributions with heavier tails, for which we instead obtain convergence to a variant of the hairy snake introduced by Janson and Marckert. We further give two applications of our main result. Firstly, we obtain a scaling limit for the difference between the height process and the Łukasiewicz path of a size-conditioned critical Bienaymé tree. Secondly, we obtain a scaling limit for the difference between the height process of a size-conditioned critical Bienaymé tree and the height process of its associated looptree.

math.PR↗
arXiv2024

A large deviation principle for the normalized excursion of $α$-stable Lévy processes without negative jumps

We establish a large deviation principle for the normalized excursion and bridge of an $α$-stable Lévy process without negative jumps, with $1<α<2$. Based on this, we derive precise asymptotics for the tail distributions of functionals of the normalized excursion and bridge, in particular, the area and maximum functionals. We advocate the use of the Skorokhod M1 topology, rather than the more usual J1 topology, as we believe it is better suited to large deviation principles for Lévy processes in general.

math.PR↗
arXiv2024

Scaling limits of random graphs

This work will appear as a chapter in a forthcoming volume titled "Topics in Probabilistic Graph Theory". A theory of scaling limits for random graphs has been developed in recent years. This theory gives access to the large-scale geometric structure of these random objects in the limit as their size goes to infinity, with distances appropriately rescaled. We start with the simplest setting of random trees, before turning to various examples of random graphs, including the critical Erdős--Rényi random graph.

math.PR↗
arXiv2021

The scaling limit of a critical random directed graph

We consider the random directed graph $\vec{G}(n,p)$ with vertex set $\{1,2,\ldots,n\}$ in which each of the $n(n-1)$ possible directed edges is present independently with probability $p$. We are interested in the strongly connected components of this directed graph. A phase transition for the emergence of a giant strongly connected component is known to occur at $p = 1/n$, with critical window $p= 1/n + λn^{-4/3}$ for $λ\in \mathcal{R}$. We show that, within this critical window, the strongly connected components of $\vec{G}(n,p)$, ranked in decreasing order of size and rescaled by $n^{-1/3}$, converge in distribution to a sequence $(\mathcal{C}_1,\mathcal{C}_2,\ldots)$ of finite strongly connected directed multigraphs with edge lengths which are either 3-regular or loops. The convergence occurs the sense of an $\ell^1$ sequence metric for which two directed multigraphs are close if there are compatible isomorphisms between their vertex and edge sets which roughly preserve the edge-lengths. Our proofs rely on a depth-first exploration of the graph which enables us to relate the strongly connected components to a particular spanning forest of the undirected Erdős-Rényi random graph $G(n,p)$, whose scaling limit is well understood. We show that the limiting sequence $(\mathcal{C}_1,\mathcal{C}_2,\ldots)$ contains only finitely many components which are not loops. If we ignore the edge lengths, any fixed finite sequence of 3-regular strongly connected directed multigraphs occurs with positive probability.

math.PR↗
arXiv2021

The stable graph: the metric space scaling limit of a critical random graph with i.i.d. power-law degrees

We prove a metric space scaling limit for a critical random graph with independent and identically distributed degrees having power-law tail behaviour with exponent $α+1$, where $α\in (1,2)$. The limiting components are constructed from random $\mathbb{R}$-trees encoded by the excursions above its running infimum of a process whose law is locally absolutely continuous with respect to that of a spectrally positive $α$-stable Lévy process. These spanning $\mathbb{R}$-trees are measure-changed $α$-stable trees. In each such $\mathbb{R}$-tree, we make a random number of vertex-identifications, whose locations are determined by an auxiliary Poisson process. This generalises results which were already known in the case where the degree distribution has a finite third moment (a model which lies in the same universality class as the Erdős--Rényi random graph) and where the role of the $α$-stable Lévy process is played by a Brownian motion.

math.PR↗
arXiv2020

Stable graphs: distributions and line-breaking construction

For $α\in (1,2]$, the $α$-stable graph arises as the universal scaling limit of critical random graphs with i.i.d. degrees having a given $α$-dependent power-law tail behavior. It consists of a sequence of compact measured metric spaces (the limiting connected components), each of which is tree-like, in the sense that it consists of an $\mathbb R$-tree with finitely many vertex-identifications (which create cycles). Indeed, given their masses and numbers of vertex-identifications, these components are independent and may be constructed from a spanning $\mathbb R$-tree, which is a biased version of the $α$-stable tree, with a certain number of leaves glued along their paths to the root. In this paper we investigate the geometric properties of such a component with given mass and number of vertex-identifications. We (1) obtain the distribution of its kernel and more generally of its discrete finite-dimensional marginals; we will observe that these distributions are related to the distributions of some configuration models (2) determine the distribution of the $α$-stable graph as a collection of $α$-stable trees glued onto its kernel and (3) present a line-breaking construction, in the same spirit as Aldous' line-breaking construction of the Brownian continuum random tree.

math.PR↗
arXiv2020

Moderate deviations of subgraph counts in the Erdős-Rényi random graphs $G(n,m)$ and $G(n,p)$

The main contribution of this article is an asymptotic expression for the rate associated with moderate deviations of subgraph counts in the Erdős-Rényi random graph $G(n,m)$. Our approach is based on applying Freedman's inequalities for the probability of deviations of martingales to a martingale representation of subgraph count deviations. In addition, we prove that subgraph count deviations of different subgraphs are all linked, via the deviations of two specific graphs, the path of length two and the triangle. We also deduce new bounds for the related $G(n,p)$ model.

math.CO↗
arXiv2019

Parking On A Random Rooted Plane Tree

In this paper, we investigate a parking process on a uniform random rooted plane tree with $n$ vertices. Every vertex of the tree has a parking space for a single car. Cars arrive at independent uniformly random vertices of the tree. If the parking space at a vertex is unoccupied when a car arrives there, it parks. If not, the car drives towards the root and parks in the first empty space it encounters (if there is one). We are interested in asymptotics of the probability of the event that all cars can park when $\lfloor αn \rfloor$ cars arrive, for $α> 0$. We observe that there is a phase transition at $α_c := \sqrt{2} -1$: if $α< α_c$ then the event has positive probability, whereas for $α> α_c$ it has probability 0. Analogous results have been proved by Lackner and Panholzer, Goldschmidt and Przykucki, and Jones for different underlying random tree models.

math.PR↗
arXiv2017

Parking on a random tree

Consider a uniform random rooted tree on vertices labelled by $[n] = \{1,2,\ldots,n\}$, with edges directed towards the root. We imagine that each node of the tree has space for a single car to park. A number $m \le n$ of cars arrive one by one, each at a node chosen independently and uniformly at random. If a car arrives at a space which is already occupied, it follows the unique path oriented towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Consider $m =[αn]$ and let $A_{n,α}$ denote the event that all $[αn]$ cars find spaces in the tree. Lackner and Panholzer proved (via analytic combinatorics methods) that there is a phase transition in this model. Then if $α\le 1/2$, we have $\mathbb{P}(A_{n,α}) \to \frac{\sqrt{1-2α}}{1-α}$, whereas if $α> 1/2$ we have $\mathbb{P}(A_{n,α}) \to 0$. We give a probabilistic explanation for this phenomenon, and an alternative proof via the objective method. Along the way, we are led to consider the following variant of the problem: take the tree to be the family tree of a Galton-Watson branching process with Poisson(1) offspring distribution, and let an independent Poisson($α$) number of cars arrive at each vertex. Let $X$ be the number of cars which visit the root of the tree. Then for $α\le 1/2$, we have $\mathbb{E}[X] \leq 1$, whereas for $α> 1/2$, we have $\mathbb{E}[X] = \infty$. This discontinuous phase transition turns out to be a generic phenomenon in settings with an arbitrary offspring distribution of mean at least 1 for the tree and arbitrary arrival distribution.

math.PR↗
arXiv2017

The spread of fire on a random multigraph

We study a model for the destruction of a random network by fire. Suppose that we are given a multigraph of minimum degree at least 2 having real-valued edge-lengths. We pick a uniform point from along the length and set it alight; the edges of the multigraph burn at speed 1. If the fire reaches a vertex of degree 2, the fire gets directly passed on to the neighbouring edge; a vertex of degree at least 3, however, passes the fire either to all of its neighbours or none, each with probability $1/2$. If the fire goes out before the whole network is burnt, we again set fire to a uniform point. We are interested in the number of fires which must be set in order to burn the whole network, and the number of points which are burnt from two different directions. We analyse these quantities for a random multigraph having $n$ vertices of degree 3 and $α(n)$ vertices of degree 4, where $α(n)/n \to 0$ as $n \to \infty$, with i.i.d. standard exponential edge-lengths. Depending on whether $α(n) \gg \sqrt{n}$ or $α(n)=O(\sqrt{n})$, we prove that as $n \to \infty$ these quantities converge jointly in distribution when suitably rescaled to either a pair of constants or to (complicated) functionals of Brownian motion. We use our analysis of this model to make progress towards a conjecture of Aronson, Frieze and Pittel concerning the number of vertices which remain unmatched when we use the Karp-Sipser algorithm to find a matching on the Erdős-Rényi random graph.

math.PR↗
arXiv2016

Inverting the cut-tree transform

We consider fragmentations of an R-tree $T$ driven by cuts arriving according to a Poisson process on $T \times [0, \infty)$, where the first co-ordinate specifies the location of the cut and the second the time at which it occurs. The genealogy of such a fragmentation is encoded by the so-called cut-tree, which was introduced by Bertoin and Miermont for a fragmentation of the Brownian continuum random tree. The cut-tree was generalised by Dieuleveut to a fragmentation of the $α$-stable trees, $α\in (1, 2)$, and by Broutin and Wang to the inhomogeneous continuum random trees of Aldous and Pitman. Remarkably, in all of these cases, the law of the cut-tree is the same as that of the original R-tree. In this paper, we develop a clean general framework for the study of cut-trees of R-trees. We then focus particularly on the problem of reconstruction: how to recover the original R-tree from its cut-tree. This has been studied in the setting of the Brownian CRT by Broutin and Wang, where they prove that it is possible to reconstruct the original tree in distribution. We describe an enrichment of the cut-tree transformation, which endows the cut tree with information we call a consistent collection of routings. We show this procedure is well-defined under minimal conditions on the R-trees. We then show that, for the case of the Brownian CRT and the $α$-stable trees with $α\in (1, 2)$, the original tree and the Poisson process of cuts thereon can both be almost surely reconstructed from the enriched cut-trees. For the latter results, our methods make essential use of the self-similarity and re-rooting invariance of these trees.

math.PR↗
arXiv2016

Behavior near the extinction time in self-similar fragmentations II: Finite dislocation measures

We study a Markovian model for the random fragmentation of an object. At each time, the state consists of a collection of blocks. Each block waits an exponential amount of time with parameter given by its size to some power $α$, independently of the other blocks. Every block then splits randomly into sub-blocks whose relative sizes are distributed according to the so-called dislocation measure. We focus here on the case where $α<0$. In this case, small blocks split intensively, and so the whole state is reduced to "dust" in a finite time, almost surely (we call this the extinction time). In this paper, we investigate how the fragmentation process behaves as it approaches its extinction time. In particular, we prove a scaling limit for the block sizes which, as a direct consequence, gives us an expression for an invariant measure for the fragmentation process. In an earlier paper [Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 338-368], we considered the same problem for another family of fragmentation processes, the so-called stable fragmentations. The results here are similar, but we emphasize that the methods used to prove them are different. Our approach in the present paper is based on Markov renewal theory and involves a somewhat unusual "spine" decomposition for the fragmentation, which may be of independent interest.

math.PR↗
arXiv2014

A line-breaking construction of the stable trees

We give a new, simple construction of the $α$-stable tree for $α\in (1,2]$. We obtain it as the closure of an increasing sequence of $\mathbb{R}$-trees inductively built by gluing together line-segments one by one. The lengths of these line-segments are related to the the increments of an increasing $\mathbb{R}_+$-valued Markov chain. For $α= 2$, we recover Aldous' line-breaking construction of the Brownian continuum random tree based on an inhomogeneous Poisson process.

math.PR↗
arXiv2013

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 tree or any rescaled version thereof. Our approach relies on a coupling between the MST problem and the Erdös-Rényi random graph. We exploit the explicit description of the scaling limit of the Erdös-Rényi random graph in the so-called critical window, established by the first three authors in an earlier paper, and provide a similar description of the scaling limit for a "critical minimum spanning forest" contained within the MST.

math.PR↗
arXiv2011

Quantum Heisenberg models and their probabilistic representations

These notes give a mathematical introduction to two seemingly unrelated topics: (i) quantum spin systems and their cycle and loop representations, due to Tóth and Aizenman-Nachtergaele; (ii) coagulation-fragmentation stochastic processes. These topics are nonetheless related, as we argue that the lengths of cycles and loops satisfy an effective coagulation-fragmentation process. This suggests that their joint distribution is Poisson-Dirichlet. These ideas are far from being proved, but they are backed by several rigorous results, notably of Dyson-Lieb-Simon and Schramm.

math-ph↗
arXiv2009

The continuum limit of critical random graphs

We consider the Erdos-Renyi random graph G(n,p) inside the critical window, that is when p=1/n+ lambda*n^{-4/3}, for some fixed lambda in R. Then, as a metric space with the graph distance rescaled by n^{-1/3}, the sequence of connected components G(n,p) converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the theory of continuum random trees. Our result gives access to the answers to a great many questions about distances in critical random graphs. In particular, we deduce that the diameter of G(n,p) rescaled by n^{-1/3} converges in distribution to an absolutely continuous random variable with finite mean.

math.PR↗