arXiv subjects

Sigurdur Örn Stefánsson

Publications and source records attributed to Sigurdur Örn Stefánsson.

11 recordsLinked to original sources

arXiv2025

The scaling limit of random 2-connected series-parallel maps

A finite graph embedded in the plane is called a series-parallel map if it can be obtained from a finite tree by repeatedly subdividing and doubling edges. We study the scaling limit of weighted random two-connected series-parallel maps with $n$ edges and show that under some integrability conditions on these weights, the maps with distances rescaled by a factor $n^{-1/2}$ converge to a constant multiple of Aldous' continuum random tree (CRT) in the Gromov--Hausdorff sense. The proof relies on a bijection between a set of trees with $n$ leaves and a set of series-parallel maps with $n$ edges, which enables one to compare geodesics in the maps and in the corresponding trees via a Markov chain argument introduced by Curien, Haas and Kortchemski (2015).

math.PR↗

arXiv2024

Physical networks as network-of-networks

Physical networks are made of nodes and links that are physical objects embedded in a geometric space. Understanding how the mutual volume exclusion between these elements affects the structure and function of physical networks calls for a suitable generalization of network theory. Here, we introduce a network-of-networks framework where we describe the shape of each extended physical node as a network embedded in space and these networks are bound together by physical links. Relying on this representation, we introduce a minimal model of network growth and we show for a general class of physical networks that volume exclusion induces heterogeneity in both node volume and degree, with the two becoming correlated. These emergent properties strongly affect the dynamics on physical networks: by calculating their Laplacian spectrum as a function of the coupling strength between the nodes we show that degree-volume correlations suppress the role of hubs as early spreaders in diffusive dynamics. We apply the network-of-networks framework to describe several real systems and find properties analog to the minimal model networks. The prevalence of these properties points towards general growth mechanisms that do not depend on the specifics of the systems.

cond-mat.stat-mech↗

arXiv2023

On scaling limits of random Halin-like maps

We consider maps which are constructed from plane trees by assigning marks to the corners of each vertex and then connecting each pair of consecutive marks on their contour by a single edge. A measure is defined on the set of such maps by assigning Boltzmann weights to the faces. When every vertex has exactly one marked corner, these maps are dissections of a polygon which are bijectively related to non-crossing trees. When every vertex has at least one marked corner, the maps are outerplanar and each of its two-connected component is bijectively related to a non-crossing tree. We study the scaling limits of the maps under these conditions and establish that for certain choices of the weights the scaling limits are either the Brownian CRT or the $α$-stable looptrees of Curien and Kortchemski.

math.PR↗

arXiv2022

Decorated stable trees

We define decorated $α$-stable trees which are informally obtained from an $α$-stable tree by blowing up its branchpoints into random metric spaces. This generalizes the $α$-stable looptrees of Curien and Kortchemski, where those metric spaces are just deterministic circles. We provide different constructions for these objects, which allows us to understand some of their geometric properties, including compactness, Hausdorff dimension and self-similarity in distribution. We prove an invariance principle which states that under some conditions, analogous discrete objects, random decorated discrete trees, converge in the scaling limit to decorated $α$-stable trees. We mention a few examples where those objects appear in the context of random trees and planar maps, and we expect them to naturally arise in many more cases.

math.PR↗

arXiv2021

Stable shredded spheres and causal random maps with large faces

We introduce a new family of random compact metric spaces $\mathcal{S}_α$ for $α\in(1,2)$, which we call stable shredded spheres. They are constructed from excursions of $α$-stable Lévy processes on $[0,1]$ possessing no negative jumps. Informally, viewing the graph of the Lévy excursion in the plane, each jump of the process is "cut open" and replaced by a circle and then all points on the graph at equal height which are not separated by a jump are identified. We show that the shredded spheres arise as scaling limits of models of causal random planar maps with large faces introduced by Di Francesco and Guitter. We also establish that their Hausdorff dimension is almost surely equal to $α$. Point identification in the shredded spheres is intimately connected to the presence of decrease points in stable spectrally positive Lévy processes as studied by Bertoin in the 90's.

math.PR↗

arXiv2017

Geometry of large Boltzmann outerplanar maps

We study the phase diagram of random outerplanar maps sampled according to non-negative Boltzmann weights that are assigned to each face of a map. We prove that for certain choices of weights the map looks like a rescaled version of its boundary when its number of vertices tends to infinity. The Boltzmann outerplanar maps are then shown to converge in the Gromov-Hausdorff sense towards the $α$-stable looptree introduced by Curien and Kortchemski (2014), with the parameter $α$ depending on the specific weight-sequence. This allows us to describe the transition of the asymptotic geometric shape from a deterministic circle to the Brownian tree.

math.PR↗

arXiv2015

Almost sure convergence of vertex degree densities in the vertex-splitting model

We study the limiting degree distribution of the vertex splitting model introduced in \cite{DDJS:2009}. This is a model of randomly growing ordered trees, where in each time step the tree is separated into two components by splitting a vertex into two, and then inserting an edge between the two new vertices. Under some assumptions on the parameters, related to the growth of the maximal degree of the tree, we prove that the vertex degree densities converge almost surely to constants which satisfy a system of equations. Using this we are also able to strengthen and prove some previously non-rigorous results mentioned in the literature.

math.PR↗

arXiv2015

Scaling limits of random planar maps with a unique large face

We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We prove that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges $n$ of the planar maps goes to infinity, the profile of distances to a marked vertex rescaled by $n^{-1/2}$ is described by a Brownian excursion. The planar maps, with the graph metric rescaled by $n^{-1/2}$, are then shown to converge in distribution toward Aldous' Brownian tree in the Gromov-Hausdorff topology. In the proofs, we rely on the Bouttier-di Francesco-Guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large.

math.PR↗

arXiv2015

Random walk on random infinite looptrees

Looptrees have recently arisen in the study of critical percolation on the uniform infinite planar triangulation. Here we consider random infinite looptrees defined as the local limit of the looptree associated with a critical Galton--Watson tree conditioned to be large. We study simple random walk on these infinite looptrees by means of providing estimates on volume and resistance growth. We prove that if the offspring distribution of the Galton--Watson process is in the domain of attraction of a stable distribution with index $α\in(1,2]$ then the spectral dimension of the looptree is $2α/(α+1)$.

math.PR↗

arXiv2015

Majority bootstrap percolation on the random graph G(n,p)

Majority bootstrap percolation on the random graph $G_{n,p}$ is a process of spread of "activation" on a given realisation of the graph with a given number of initially active nodes. At each step those vertices which have more active neighbours than inactive neighbours become active as well. We study the size $A^*$ of the final active set. The parameters of the model are, besides $n$ (tending to $\infty$), the size $A(0)=A_0(n)$ of the initially active set and the probability $p=p(n)$ of the edges in the graph. We prove that the process cannot percolate for $A(0) = o(n)$. We study the process for $A(0) = θn$ and every range of $p$ and show that the model exhibits different behaviours for different ranges of $p$. For very small $p \ll \frac{1}{n}$, the activation does not spread significantly. For large $p \gg \frac{1}{n}$ then we see a phase transition at $A(0) \simeq \frac{1}{2}n$. In the case $p= \frac{c}{n}$, the activation propagates to a significantly larger part of the graph but (the process does not percolate) a positive part of the graph remains inactive.

math.PR↗

arXiv2015

On site percolation in random quadrangulations of the half-plane

We study site percolation on uniform quadrangulations of the upper half plane. The main contribution is a method for applying Angel's peeling process, in particular for analyzing an evolving boundary condition during the peeling. Our method lets us obtain rigorous and explicit upper and lower bounds on the percolation threshold $p_\mathrm{c}$, and thus show in particular that $0.5511\leq p_\mathrm{c}\leq 0.5581$. The method can be extended to site percolation on other half-planar maps with the domain Markov property.

math.PR↗