arXiv subjects

Frederik Ravn Klausen

Publications and source records attributed to Frederik Ravn Klausen.

At least 19 recordsLinked to original sources

arXiv2026

Ensuring proportionality: a logical model for compensatory seats added to multi-member constituencies

How many compensatory seats are needed to ensure (full) proportionality in two-tier electoral systems with multi-member constituencies? This question is answered by estimating the discrepancy between seats and votes rigorously, probabilistically and empirically. In 1919, Pólya showed that the Jefferson/D'Hondt method favours large parties. This seat surplus accumulates across constituencies. In some countries (e.g., Portugal, Poland, Spain and Turkey), this accumulation directly leads to quantifiable disproportionality in the final parliament. Elsewhere compensatory seats counteract this accumulated seat surplus. Using a logic-first approach, closed formulas for the required number of compensatory seats are derived given regional and national apportionment methods, the number of relevant parties $n$ and the size of the largest party $p$. If there are $c$ constituencies, it is derived logically that a two-tier system built on local D'Hondt apportionment in multi-member constituencies needs around $\frac{c}{2}(n-\frac{1}{p})$ compensatory seats to ensure proportionality. In words, that is half a compensatory seat per constituency times a number that increases with the number of parties and the size of the largest party. The validity of the derived formulas is analysed empirically using the extensive recent CLEA dataset. Many quantities do not depend much on the total parliament sizes, which means that the variance can be reduced by artificially varying the house sizes. Portugal, Poland, Spain and Turkey all employ a one-tier D'Hondt system and the analysis shows that if they were to add compensatory seats to the system, reasonable numbers would be 40, 75, 75 and 120, assuming that no other properties of the systems change.

physics.soc-ph↗

arXiv2026

A simple proof of exponential decay for the near-critical planar Ising model

For the Ising model defined on $a\mathbb{Z}^2$ at critical temperature with external field $a^{15/8}h$, we give a simple and elementary proof that its truncated two-point function decays exponentially. The proof combines the high temperature expansion, random-cluster and random current representations. A new input in the proof is that, in the near-critical sourceless single current measure, there are many loops formed by a path on $a\mathbb{Z}^2$ with diameter of order $1$, together with two external edges that connect the path's endpoints to the ghost.

math.PR↗

arXiv2026

The Supercritical Loop O(1) and Random Current models: Uniqueness and Mixing

Much recent rigorous study of the classical ferromagnetic Ising model has been powered by its graphical representations, such as the random current and loop O(1) model (high temperature expansion). In this paper, we prove uniqueness of Gibbs measures and exponential ratio weak mixing for the loop O(1) and random current models corresponding to the supercritical Ising model on the hypercubic lattice $\Z^d$ in any dimension $d \geq 2$. The methods generalise to $q$-flow models and have natural applications for gradient measures of $\Z/q\mathbb{Z}$-gauge theories.

math.PR↗

arXiv2026

Half-plane non-coexistence without FKG

For $μ$ an edge percolation measure on the infinite square lattice, let $μ_{\textit{hp}}$ (respectively, $μ^*_{hp}$) denote its marginal (respectively, the marginal of its planar dual process) on the upper half-plane. We show that if $μ$ is translation-invariant and ergodic and almost surely has only finitely many infinite clusters, then either almost surely $μ_{hp}$ has no infinite cluster, or almost surely $μ^*_{hp}$ has no infinite cluster. By the classical Burton--Keane argument, these hypotheses are satisfied if $μ$ is translation-invariant and ergodic and has finite-energy. In contrast to previous ``non-coexistence'' theorems, our result does not impose a positive-correlation (FKG) hypothesis on $μ$. Our arguments also apply to the random-cluster model (including the regime $q<1$, which lacks FKG), the uniform spanning tree, and the uniform odd subgraph.

math.PR↗

arXiv2026

A General Coupling for Ising Models and Beyond

The couplings between the Ising model and its graphical representations, the random-cluster, random current and loop $\mathrm{O}(1)$ models, are put on common footing through a generalization of the Swendsen-Wang-Edwards-Sokal coupling. A new special case is that the single random current with parameter $2J$ on edges of agreement of XOR-Ising spins has the law of the double random current. The coupling also yields a general mechanism for constructing conditional Bernoulli percolation measures via uniform sampling, providing new perspectives on models such as the arboreal gas, random $d$-regular graphs, and self-avoiding walks. As an application of the coupling for the Loop-Cluster joint model, we prove that the regimes of exponential decay of the $q$-state Potts model, the random-cluster representation, and its $q$-flow loop representation coincide on the torus, generalizing the Ising result. By providing a new and equivalent definition of the plaquette random cluster model, we are able to generalize the loop-cluster coupling to lattice gauge theories.

math.PR↗

arXiv2026

From Kreweras walks to branching perimeter processes of percolated triangulations

In this note, using a result of Bernardi, Holden and Sun, we give an explicit geometric relation between the perimeter of the peeling process along the percolation interface of a triangulation, and the corresponding Kreweras walk. The relation naturally extends to the branching peeling exploration. This sheds light on the relation between the growth-fragmentation process and correlated Brownian excursions discovered by Da Silva, Powell and Watson.

math.CO↗

arXiv2026

The End of the Double Random Current

The Ising correlation function can be expressed in terms of connectivity probabilities of a random graph known as the double random current. In 2016, Duminil-Copin asked whether this random graph only has one topological end. One-endedness implies that free and wired random-cluster measures are equal and that any translation-invariant Gibbs measure of the Ising model is a convex combination of the $+$ and $-$ Gibbs measures. In the general setting of transitive amenable graphs, these properties have been established by Raoufi, effectively circumventing the question of one-endedness. In this paper, we prove one-endedness of the double random current on any transitive, amenable one-ended graph and, relying on Raoufi's work, we answer the question of Duminil-Copin in the affirmative in this general setup. The results of this paper do not give a new proof of uniqueness of random-cluster measure or the characterisations of Gibbs states for the Ising model.

math.PR↗

arXiv2026

Powers in prime bases and a problem on central binomial coefficients

It is an open problem whether $ \binom{2n}{n} $ is divisible by 4 or 9 for all $n>256$. In connection with this, we prove that for a fixed uneven $m$ the asymptotic density of $k$'s such that $ m \nmid \binom{2^{k+1}}{2^{k}} $ is 0. To do so we examine numbers of the form $α^{k}$ in base $p$, where $p$ is a prime and $(α, p)=1$. For every $n$ and $a$ we find an upper bound on the number of $k$'s less than $a$ such that $(α^{k})_p$ contains less than $n$ digits greater than $\frac{p}{2}$. This is done by showing that every sequence of the form $\langle σ_t, \dots, σ_1,σ_0 \rangle$, where $0\leq σ_i<p$ for $i\geq 1$ and $σ_0$ is in the residue class generated by $α$ modulo $p$, occurs at specific places in the representation $(α^k)_p$ as $k$ varies.

math.NT↗

arXiv2025

Non-uniqueness of phase transitions for graphical representations of the Ising model on tree-like graphs

We consider the graphical representations of the Ising model on tree-like graphs. We construct a class of graphs on which the loop $\mathrm{O}(1)$ model and the single random current exhibit a non-unique phase transition with respect to the inverse temperature, highlighting the non-monotonicity of both models. It follows from the construction that there exist infinite graphs $\mathbb{G}\subseteq \mathbb{G}'$ such that the uniform even subgraph of $\mathbb{G}'$ percolates and the uniform even subgraph of $\mathbb{G}$ does not. We also show that on the wired $d$-regular tree, the phase transitions of the loop $\mathrm{O}(1)$, the single random current, and the random-cluster models are all unique and coincide.

math.PR↗

arXiv2025

Spectra of generators of Markovian evolution in the thermodynamic limit: From non-Hermitian to full evolution via tridiagonal Laurent matrices

It is shown that generators of single-particle, translation-invariant Lindblad operators on the infinite line are unitarily equivalent to direct integrals of finite-range bi-infinite Laurent operator with finite-range perturbations. This representation enables rigorous calculation of spectra for local dissipation such as dephasing and incoherent hopping, and yields proofs of gaplessness, absence of residual spectrum and a condition for convergence of finite volume spectra to their infinite volume counterparts. The analysis relies on new results on the spectra of direct integrals of non-normal operators which may be of independent interest.

math-ph↗

arXiv2025

Fault-tolerant quantum simulation of generalized Hubbard models

Quantum simulations of strongly interacting fermionic systems, such as those described by the Hubbard model, are promising candidates for useful early fault-tolerant quantum computing applications. This paper presents Tile Trotterization, a generalization of plaquette Trotterization (PLAQ), which uses a set of tiles to construct Trotter decompositions of arbitrary lattice Hubbard models. The Tile Trotterization scheme also enables the simulation of more complex models, including the extended Hubbard model. We improve previous Hubbard model commutator bounds, further provide tight commutator bounds for periodic extended Hubbard models, and demonstrate the use of tensor network methods for this task. We consider applications of Tile Trotterization to simulate hexagonal lattice Hubbard models and compare the resource requirements of Tile Trotterization for performing quantum phase estimation to a qubitization-based approach, demonstrating that Tile Trotterization scales more efficiently with system size. These advancements significantly broaden the potential applications of early fault-tolerant quantum computers to models of practical interest in materials research and organic chemistry.

quant-ph↗

arXiv2025

The Uniform Even Subgraph and Its Connection to Phase Transitions of Graphical Representations of the Ising Model

The uniform even subgraph is intimately related to the Ising model, the random-cluster model, the random current model, and the loop $\mathrm{O}$(1) model. In this paper, we first prove that the uniform even subgraph of $Z^d$ percolates for $d \geq 2$ using its characterisation as the Haar measure on the group of even graphs. We then tighten the result by showing that the loop $\mathrm{O}$(1) model on $Z^d$ percolates for $d \geq 2$ for edge-weights $x$ lying in some interval $(1-\varepsilon,1]$. Finally, our main theorem is that the loop $\mathrm{O}$(1) model and random current models corresponding to a supercritical Ising model are always at least critical, in the sense that their two-point correlation functions decay at most polynomially and the expected cluster sizes are infinite.

math.PR↗

arXiv2024

Going Beyond Gadgets: The Importance of Scalability for Analogue Quantum Simulators

Quantum hardware has the potential to efficiently solve computationally difficult problems in physics and chemistry to reap enormous practical rewards. Analogue quantum simulation accomplishes this by using the dynamics of a controlled many-body system to mimic those of another system; such a method is feasible on near-term devices. We show that previous theoretical approaches to analogue quantum simulation suffer from fundamental barriers which prohibit scalable experimental implementation. By introducing a new mathematical framework and going beyond the usual toolbox of Hamiltonian complexity theory with an additional resource of engineered dissipation, we show that these barriers can be overcome. This provides a powerful new perspective for the rigorous study of analogue quantum simulators.

quant-ph↗

arXiv2024

Decoherence is an echo of Anderson localization in open quantum systems

We study the time evolution of single-particle quantum states described by a Lindblad master equation with local terms. By means of a geometric resolvent equation derived for Lindblad generators, we establish a finite-volume-type criterion for the decay of the off-diagonal matrix elements in the position basis of the time-evolved or steady states. This criterion is shown to yield exponential decay for systems where the non-hermitian evolution is either gapped or strongly disordered. The gap exists for example whenever any level of local dephasing is present in the system. The result in the disordered case can be viewed as an extension of Anderson localization to open quantum systems.

math-ph↗

arXiv2023

Random Problems in Mathematical Physics

This PhD thesis deals with a number of different problems in mathematical physics with the common thread that they have probabilistic aspects. The problems all stem from mathematical studies of lattice systems in statistical and quantum physics; however beyond that, the selection of the concrete problems is to a certain extent arbitrary. This thesis consists of an introduction and seven papers.

math-ph↗

arXiv2023

Stochastic cellular automaton model of culture formation

We introduce a stochastic cellular automaton as a model for culture and border formation. The model can be conceptualized as a game where the expansion rate of cultures is quantified in terms of their area and perimeter in such a way that approximately geometrically round cultures get a competitive advantage. We first analyse the model with periodic boundary conditions, where we study how the model can end up in a fixed state, i.e. freezes. Then we implement the model on the European geography with mountains and rivers. We see how the model reproduces some qualitative features of European culture formation, namely that rivers and mountains are more frequently borders between cultures, mountainous regions tend to have higher cultural diversity and the central European plain has less clear cultural borders.

physics.soc-ph↗

arXiv2023

Strict monotonicity, continuity and bounds on the Kertész line for the random-cluster model on $\mathbb{Z}^d$

Ising and Potts models can be studied using the Fortuin-Kasteleyn representation through the Edwards-Sokal coupling. This adapts to the setting where the models are exposed to an external field of strength $h>0$. In this representation, which is also known as the random-cluster model, the Kertész line separates the two regions of parameter space according to the existence of an infinite cluster in $\mathbb{Z}^d$. This signifies a geometric phase transition between the ordered and disordered phases even in cases where a thermodynamic phase transition does not occur. In this article, we prove strict monotonicity and continuity of the Kertész line. Furthermore, we give new rigorous bounds that are asymptotically correct in the limit $h \to 0$ complementing the bounds from the work of Ruiz and Wouts [J. Math. Phys. 49, 053303 (2008)], which were asymptotically correct for $h \to \infty$. Finally, using a cluster expansion, we investigate the continuity of the Kertész line phase transition.

math-ph↗

arXiv2022

Mass scaling of the near-critical 2D Ising model using random currents

We examine the Ising model at its critical temperature with an external magnetic field $h a^{\frac{15}{8}}$ on $a\mathbb{Z}^2$ for $a,h >0$. A new proof of exponential decay of the truncated two-point correlation functions is presented. It is proven that the mass (inverse correlation length) is of the order of $h^\frac{8}{15}$ in the limit $h \to 0$. This was previously proven with CLE-methods in $\lbrack 1 \rbrack$. Our new proof uses instead the random current representation of the Ising model and its backbone exploration. The method further relies on recent couplings to the random cluster model $\lbrack 2 \rbrack$ as well as a near-critical RSW-result for the random cluster model $\lbrack 3 \rbrack$.

math-ph↗