arXiv subjects

Jakob Björnberg

Publications and source records attributed to Jakob Björnberg.

3 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↗

arXiv2022

Heisenberg models and Schur--Weyl duality

We present a detailed analysis of certain quantum spin systems with inhomogeneous (non-random) mean-field interactions. Examples include, but are not limited to, the interchange- and spin singlet projection interactions on complete bipartite graphs. Using two instances of the representation theoretic framework of Schur--Weyl duality, we can explicitly compute the free energy and other thermodynamic limits in the models we consider. This allows us to describe the phase-transition, the ground-state phase diagram, and the expected structure of extremal states.

math-ph↗

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↗