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Jakob E. Björnberg

Publications and source records attributed to Jakob E. Björnberg.

At least 19 recordsLinked to original sources
arXiv2026

A random recursive tree model with doubling events

We introduce a new model of random tree that grows like a random recursive tree, except at some exceptional "doubling events" when the tree is replaced by two copies of itself attached to a new root. We prove asymptotic results for the size of this tree at large times, its degree distribution, and its height profile. We also prove a lower bound for its height. Because of the doubling events that affect the tree globally, the proofs are all much more intricate than in the case of the random recursive tree in which the growing operation is always local.

math.PR↗
arXiv2025

Exponential decay in $O(n)$-invariant quantum spin systems

We consider $O(n)$-invariant and reflection-positive quantum spin systems on the integer lattice in any dimension, and prove that spin-spin correlations decay exponentially fast provided n is large enough. This answers a question of Ueltschi, who proved that for small n there is instead long-range order (for d at least 3).

math-ph↗
arXiv2024

A two-table theorem for a disordered Chinese restaurant process

We investigate a disordered variant of Pitman's Chinese restaurant process where tables carry i.i.d. weights. Incoming customers choose to sit at an occupied table with a probability proportional to the product of its occupancy and its weight, or they sit at an unoccupied table with a probability proportional to a parameter $θ>0$. This is a system out of equilibrium where the proportion of customers at any given table converges to zero almost surely. We show that for weight distributions in any of the three extreme value classes, Weibull, Gumbel or Fréchet, the proportion of customers sitting at the largest table converges to one in probability, but not almost surely, and the proportion of customers sitting at either of the largest two tables converges to one almost surely.

math.PR↗
arXiv2022

Correlation inequalities for the uniform 8-vertex model and the toric code model

We elucidate connections between four models in statistical physics and probability theory: (1) the toric code model of Kitaev, (2) the uniform eight-vertex model, (3) random walk on a hypercube, and (4) a classical Ising model with four-body interaction. As a consequence of our analysis (and of the GKS-inequalities for the Ising model) we obtain correlation inequalities for the toric code model and the uniform eight-vertex model.

math.PR↗
arXiv2021

Dimerization in quantum spin chains with $O(n)$ symmetry

We consider quantum spins with $S\geq1$, and two-body interactions with $O(2S+1)$ symmetry. We discuss the ground state phase diagram of the one-dimensional system. We give a rigorous proof of dimerization for an open region of the phase diagram, for $S$ sufficiently large. We also prove the existence of a gap for excitations.

math-ph↗
arXiv2019

Quantum spins and random loops on the complete graph

We present a systematic analysis of quantum Heisenberg-, XY- and interchange models on the complete graph. These models exhibit phase transitions accompanied by spontaneous symmetry breaking, which we study by calculating the generating function of expectations of powers of the averaged spin density. Various critical exponents are determined. Certain objects of the associated loop models are shown to have properties of Poisson--Dirichlet distributions.

math-ph↗
arXiv2019

Characterising random partitions by random colouring

Let $(X_1,X_2,...)$ be a random partition of the unit interval $[0,1]$, i.e. $X_i\geq0$ and $\sum_{i\geq1} X_i=1$, and let $(\varepsilon_1,\varepsilon_2,...)$ be i.i.d. Bernoulli random variables of parameter $p \in (0,1)$. The Bernoulli convolution of the partition is the random variable $Z =\sum_{i\geq1} \varepsilon_i X_i$. The question addressed in this article is: Knowing the distribution of $Z$ for some fixed $p\in(0,1)$, what can we infer about the random partition? We consider random partitions formed by residual allocation and prove that their distributions are fully characterised by their Bernoulli convolution if and only if the parameter $p$ is not equal to $1/2$.

math.PR↗
arXiv2019

The interchange process with reversals on the complete graph

We consider an extension of the interchange process on the complete graph, in which a fraction of the transpositions are replaced by `reversals'. The model is motivated by statistical physics, where it plays a role in stochastic representations of $XXZ$-models. We prove convergence to PD($\tfrac12$) of the rescaled cycle sizes, above the critical point for the appearance of macroscopic cycles. This extends a result of Schramm on convergence to PD(1) for the usual interchange process.

math.PR↗
arXiv2018

Critical temperature of Heisenberg models on regular trees, via random loops

We estimate the critical temperature of a family of quantum spin systems on regular trees of large degree. The systems include the spin-$\frac12$ XXZ model and the spin-1 nematic model. Our formula is conjectured to be valid for large-dimensional cubic lattices. Our method of proof uses a probabilistic representation in terms of random loops.

math-ph↗
arXiv2017

Critical parameter of random loop model on trees

We give estimates of the critical parameter for random loop models that are related to quantum spin systems. A special case of the model that we consider is the interchange- or random-stirring process. We consider here the model defined on regular trees of large degrees, which are expected to approximate high spatial dimensions. We find a critical parameter that indeed shares similarity with existing numerical results for the cubic lattice. In the case of the interchange process our results improve on earlier work by Angel and by Hammond, in that we determine the second-order term of the critical parameter.

math.PR↗
arXiv2016

Fermionic observables in the transverse Ising chain

We introduce a notion of s-holomorphicity suitable for certain quantum spin systems in one dimension, and define two observables in the critical transverse-field Ising model which have this property. The observables are defined using graphical representations in the complex plane, and are analogous to Smirnov's FK-Ising and spin-Ising observables, respectively. We also briefly discuss scaling-limits of these observables.

math-ph↗
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

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↗
arXiv2015

Decay of transverse correlations in quantum Heisenberg models

We study a class of quantum spin systems that includes the $S=\tfrac12$ Heisenberg and XY-models, and prove that two-point correlations exhibit exponential decay in the presence of a transverse magnetic field. The field is not necessarily constant, it may be random, and it points in the same direction. Our proof is entirely probabilistic and it relies on a random-loop-representation of the correlation functions, on stochastic domination, and on first-passage percolation.

math-ph↗
arXiv2014

Vanishing critical magnetization in the quantum Ising model

Adapting the recent argument of Aizenman, Duminil-Copin and Sidoravicius for the classical Ising model, it is shown here that the magnetization in the transverse-field Ising model vanishes at the critical point. The proof applies to the ground state in dimension $d\geq2$ and to positive-temperature states in dimension $d\geq 3$, and relies on graphical representations as well as an infrared bound.

math-ph↗
arXiv2014

Recurrence of bipartite planar maps

This paper concerns random bipartite planar maps which are defined by assigning weights to their faces. The paper presents a threefold contribution to the theory. Firstly, we prove the existence of the local limit for all choices of weights and describe it in terms of an infinite mobile. Secondly, we show that the local limit is in all cases almost surely recurrent. And thirdly, we show that for certain choices of weights the local limit has exactly one face of infinite degree and has in that case spectral dimension $4/3$ (the latter requires a mild moment condition).

math.PR↗
arXiv2013

Infrared bound and mean-field behaviour in the quantum Ising model

We prove an infrared bound for the transverse field Ising model. This bound is stronger than the previously known infrared bound for the model, and allows us to investigate mean-field behaviour. As an application we show that the critical exponent $γ$ for the susceptibility attains its mean-field value $γ=1$ in dimension at least 4 (positive temperature), respectively 3 (ground state), with logarithmic corrections in the boundary cases.

math.PR↗