arXiv subjects

Malin Palö Forsström

Publications and source records attributed to Malin Palö Forsström.

At least 19 recordsLinked to original sources

arXiv2026

Poisson representations for tree-indexed Markov chains

In~\cite{fgs}, the class of Poisson representable processes was introduced. Several well-known processes were shown not to belong to this class, with examples including both the Curie Weiss model and the Ising model on $ \mathbb{Z}^2 $ for certain choices of parameters. Curiously, it was also shown that all positively associated $ \{ 0,1 \}$-valued Markov chains do belong to this class. In this paper, we interpolate between Markov chains and Ising models by considering tree-indexed Markov chains. In particular, we show that for any finite tree that is not a path, whether or not the corresponding tree-indexed Markov chain is representable always depends on the parameters. Moreover, we give an example of a family of infinite trees such that the corresponding tree-indexed Markov chains are representable for some non-trivial parameters. In addition, we give alternative proofs and arguments and also strengthen several of the results in~\cite{fgs}.

math.PR↗

arXiv2026

Phase transitions in the charged compact abelian lattice Higgs model

We consider the (compact) abelian lattice Higgs model with charge $ k \geq 1 $ and show, using charged Wilson~loop observables and charged versions of the Marcu--Fredenhagen ratio, that this model exhibits several distinct phase transitions. In particular, we show that if $k = 2$, then the Marcu--Fredenhagen ratio and Wilson~loop observables together can distinguish among three distinct phases of the parameter space, and hence both can be used as order parameters for the model.

math.PR↗

arXiv2025

Ornstein-Zernike decay of Wilson line observables in the free phase of the $ \mathbb{Z}_2 $ lattice Higgs model

In the physics literature, the Wilson line observable is believed to have a phase transition between a region with pure exponential decay and a region with Ornstein-Zernike type corrections. In~\cite{f2024b}, we confirmed the first part of this prediction. In this paper, we complement these results by showing that if $ κ$ is small and $ β$ large compared to the length of the line, then Wilson line expectations have exponential decay with Ornstein-Zernike type behavior.

math.PR↗

arXiv2025

Mixing for Poisson representable processes and consequences for the Ising model and the contact process

Forsström et al. [8] recently introduced a large class of $\{0,1\}$-valued processes that they named Poisson representable. In addition to deriving several interesting properties for these processes, their main focus was determining which processes are contained in this class. In this paper, we derive new characteristics for Poisson representable processes in terms of certain mixing properties. Using these, we argue that neither the upper invariant measure of the supercritical contact process on $\mathbb{Z}^d$ nor the plus state of the Ising model on $\mathbb{Z}^2$ within the phase transition regime is Poisson representable. Moreover, we show that on $\mathbb{Z}^d$, $d\geq 2$, any non-extremal translation invariant state of the Ising model cannot be Poisson representable. Together, these results provide answers to questions raised in [8].

math.PR↗

arXiv2024

Wilson lines in the Abelian lattice Higgs model

Lattice gauge theories are lattice approximations of the Yang-Mills theory in physics. The abelian lattice Higgs model is one of the simplest examples of a lattice gauge theory interacting with an external field. In a previous paper~\cite{flv2021}, we calculated the leading order term of the expected value of Wilson loop observables in the low-temperature regime of the abelian lattice Higgs model on $ \mathbb{Z}^4 ,$ with structure group $G = \mathbb{Z}_n $ for some $ n \geq 2. $ In the absence of a Higgs field, these are important observables since they exhibit a phase transition which can be interpreted as distinguishing between regions with and without quark confinement. However, in the presence of a Higgs field, this is no longer the case, and a more relevant family of observables are so-called open Wilson lines. In this paper, we extend and refine the ideas introduced in~\cite{flv2021} to calculate the leading order term of the expected value of the more general Wilson line observables. Using our main result, we then calculate the leading order term of several natural ratios of expected values and confirm the behavior predicted by physicists.

math.PR↗

arXiv2023

Free energy and quark potential in Ising lattice gauge theory via cluster expansion

We revisit the cluster expansion for Ising lattice gauge theory on $\mathbb{Z}^m, \, m \ge 3,$ with Wilson action, at a fixed inverse temperature $ β$ in the low-temperature regime. We prove existence and analyticity of the infinite volume limit of the free energy and compute the first few terms in its expansion in powers of $e^{-β}$. We further analyze Wilson loop expectations and derive an estimate that shows how the lattice scale geometry of a loop is reflected in the large $β$ asymptotic expansion. Specializing to axis parallel rectangular loops $γ_{T,R}$ with side-lengths $T$ and $R$, we consider the limiting function $$ V_β(R) := \lim_{T \to \infty} - \frac{1}{T} \log \, \langle W_{γ_{T,R}} \rangle_β, $$ known as the static quark potential in the physics literature. We verify existence of the limit (with an estimate on the convergence rate) and compute the first few terms in the expansion in powers of $e^{-β}$. As a consequence, a strong version of the perimeter law follows. We also treat $- \log \, \langle W_{γ_{T,R}} \rangle_β/ (T+R)$ as $T, R$ tend to infinity simultaneously and give analogous estimates.

math.PR↗

arXiv2021

Wilson loops in finite Abelian lattice gauge theories

We consider lattice gauge theories on $\mathbb{Z}^4$ with Wilson action and structure group $\mathbb{Z}_n$. We compute the expectation of Wilson loop observables to leading order in the weak coupling regime, extending and refining a recent result of Chatterjee. Our proofs use neither duality relations nor cluster expansion techniques.

math.PR↗

arXiv2021

When are sequences of Boolean functions tame?

In \cite{js2006}, Jonasson and Steif conjectured that no non-degenerate sequence of transitive Boolean functions $ (f_n)_{n \geq 1}$ with $ \lim_{n \to \infty} I(f_n)= \infty $ could be tame (with respect to some $ (p_n)_{n \geq 1} $). In a companion paper \cite{f}, the author showed that this conjecture in its full generality is false, by providing a counter-example for the case when, at the same time, $\lim_{n \to \infty} np_n = \infty $ and $ \lim_{n \to \infty} n^αp_n = 0$ for some $ α\in (0,1 ).$ In this paper we show that with slightly different assumptions, the conclusion of the conjecture holds when the sequence $(p_n)_{n \geq 1}$ is bounded away from zero and one.

math.PR↗

arXiv2021

Decay of correlations in finite Abelian lattice gauge theories

In this paper, we study lattice gauge theory on $ \mathbb{Z}^4 $ with finite Abelian structure group. When the inverse coupling strength is sufficiently large, we give an upper bound on the decay of correlations of local functions and compute the leading order term for both the expected value of the spin at a given plaquette as well as for the two-point correlation function. Moreover, we give an upper bound on the dependency of the size of the box on which the model is defined. The results in this paper extend and refine results by Chatterjee and Borgs.

math.PR↗

arXiv2020

A tame sequence of transitive Boolean functions

Given a sequence of Boolean functions $(f_n)_{n \geq 1}$, $f_n \colon \{ 0,1 \}^{n} \to \{ 0,1 \}$, and a sequence $(X^{(n)})_{n\geq 1} $ of continuous time $p_n $-biased random walks $ X^{(n)} = (X_t^{(n)})_{t \geq 0}$ on $ \{ 0,1 \}^{n}$, let $ C_n $ be the (random) number of times in $(0,1) $ at which the process $ (f_n(X_t))_{t \geq 0} $ changes its value. In \cite{js2006}, the authors conjectured that if $ (f_n)_{n \geq 1} $ is non-degenerate, transitive and satisfies $ \lim_{n \to \infty} \mathbb{E}[C_n] = \infty$, then $ (C_n)_{n \geq 1} $ is not tight. We give an explicit example of a sequence of Boolean functions which disproves this conjecture.

math.PR↗

arXiv2020

Color representations of Ising models

In \cite{st2017}, the authors introduced the so-called \emph{general divide and color models}. One of the most well-known examples of such a model is the Ising model with external field $h = 0$, which has a color representation given by the random cluster model. In this paper, we give necessary and sufficient conditions for this color representation to be unique. We also show that if one considers the Ising model on a complete graph, then for many $h > 0$, there is no color representation. This shows in particular that any generalization of the random cluster model which provides color representations of Ising models with external fields cannot, in general, be a generalized divide and color models. Furthermore, we show that there can be at most finitely many $β> 0$ at which the random cluster model can be continuously extended to a color representation for $h \not = 0$.

math.PR↗

arXiv2020

A formula for hidden regular variation behavior for symmetric stable distributions

We develop a formula for the power-law decay of various sets for symmetric stable random vectors in terms of how many vectors from the support of the corresponding spectral measure are needed to enter the set. One sees different decay rates in "different directions", illustrating the phenomenon of hidden regular variation. We give several examples and obtain quite varied behavior, including sets which do not have exact power-law decay.

math.PR↗

arXiv2020

Divide and color representations for threshold Gaussian and stable vectors

We study the question of when a (\{0,1\})-valued threshold process associated to a mean zero Gaussian or a symmetric stable vector corresponds to a {\it divide and color (DC) process}. This means that the process corresponding to fixing a threshold level $h$ and letting a 1 correspond to the variable being larger than $h$ arises from a random partition of the index set followed by coloring {\it all} elements in each partition element 1 or 0 with probabilities $p$ and $1-p$, independently for different partition elements. While it turns out that all discrete Gaussian free fields yield a DC process when the threshold is zero, for general $n$-dimensional mean zero, variance one Gaussian vectors with nonnegative covariances, this is true in general when $n=3$ but is false for $n=4$. The behavior is quite different depending on whether the threshold level $h$ is zero or not and we show that there is no general monotonicity in $h$ in either direction. We also show that all constant variance discrete Gaussian free fields with a finite number of variables yield DC processes for large thresholds. In the stable case, for the simplest nontrivial symmetric stable vector with three variables, we obtain a phase transition in the stability exponent $α$ at the surprising value of $1/2$; if the index of stability is larger than $1/2$, then the process yields a DC process for large $h$ while if the index of stability is smaller than $1/2$, then this is not the case.

math.PR↗

arXiv2020

An analysis of the induced linear operators associated to divide and color models

We study the natural linear operators associated to divide and color (DC) models. The degree of nonuniqueness of the random partition yielding a DC model is directly related to the dimension of the kernel of these linear operators. We determine exactly the dimension of these kernels as well as analyze a permutation-invariant version. We also obtain properties of the solution set for certain parameter values which will be important in (1) showing that large threshold discrete Gaussian free fields are DC models and in (2) analyzing when the Ising model with a positive external field is a DC model, both in future work. However, even here, we give an application to the Ising model on a triangle.

math.PR↗

arXiv2019

A Few Surprising Integrals

Using formulas for certain quantities involving stable vectors, due to I. Molchanov, and in some cases utilizing the so-called divide and color model, we prove that certain families of integrals which, ostensibly, depend on a parameter are in fact independent of this parameter.

math.PR↗

arXiv2018

Denseness of volatile and nonvolatile sequences of functions

In a recent paper by Jonasson and Steif, definitions to describe the volatility of sequences of Boolean functions, $ f_n \colon \{ -1,1 \}^n \to \{ -1,1 \} $ were introduced. We continue their study of how these definitions relate to noise stability and noise sensitivity. Our main results are that the set of volatile sequences of Boolean functions is a natural way "dense" in the set of all sequences of Boolean functions, and that the set of non-volatile Boolean sequences is not "dense" in the set of noise stable sequences of Boolean functions.

math.PR↗

arXiv2017

Exact Hausdorff measures of Cantor sets

Cantor sets in $\mathbb{R}$ are common examples of sets for which Hausdorff measures can be positive and finite. However, there exist Cantor sets for which no Hausdorff measure is supported and finite. The purpose of this paper is to try to resolve this problem by studying an extension of the Hausdorff measures $ μ_h$ on $\mathbb{R}$, allowing gauge functions to depend on the midpoint of the covering intervals instead of only on the diameter. As a main result, a theorem about the Hausdorff measure of any regular enough Cantor set, with respect to a chosen gauge function, is obtained.

math.MG↗

arXiv2016

Monotonicity properties of exclusion sensitivity

In~\cite{bgs2013}, exclusion sensitivity and exclusion stability for symmetric exclusion processes on graphs were defined as a natural analogue of noise sensitivity and noise stability in this setting. As these concepts were defined for any sequence of connected graphs, it is natural to study the monotonicity properties of these definitions with respect to adding edges to the graphs, and in particular, whether some graphs are more stable or sensitive than others. The main purpose of this paper is to answer some such question from~\cite{bgs2013}. The main tool used is included results about the eigenvectors and eigenvalues of the generator of symmetric exclusion processes on complete graphs.

math.PR↗