arXiv subjects

Andreas Klippel

Publications and source records attributed to Andreas Klippel.

9 recordsLinked to original sources

arXiv2026

Dimer model and random lattice permutations with general weights

The dimer model and random lattice permutations are two fundamental objects at the interface of probability, combinatorics, and mathematical physics. We study these models on finite periodic boxes in $\mathbb Z^d$ within a common framework. For the dimer model, edges connecting arbitrary vertices carry a weight which depends on their relative displacement and dimer configurations are weighted through their occupied edges. Superimposing two independent perfect matchings gives the double-dimer model, whose configurations are collections of disjoint loops. Permutations, instead, are weighted through the spatial displacement of their jumps. For broad classes of weights of finite or infinite range we prove long-range order and the occurrence of macroscopic loops. This extends nearest-neighbour results to arbitrary-range edges and jumps. In particular, long-range weights yield long-range order and macroscopic loops already in dimensions $d=1,2$. In dimension two, this behaviour is qualitatively different from that of the nearest-neighbour model. We complement these results with sharp absence criteria.

math.PR↗

arXiv2026

Critical-curve regularity for finite-lifespan frog models via local-to-global comparisons

We study the phase boundary of the finite-lifespan frog model. For rate-one continuous-time simple random walk on an infinite, connected, locally finite transitive graph of superlinear growth, we prove that the critical-density curve is continuous and strictly decreasing, with $-\logλ_c$ locally bi-Lipschitz. The inverse critical-lifespan curve has the corresponding regularity wherever it is finite. Whenever every positive density has finite critical lifespan, this resolves a conjecture of Angel, de la Riva, Hermon, and Shi on the regularity of the critical-parameter curves; in particular, it does so on nonamenable and superlinear polynomial-growth graphs. We also establish a small-lifespan scaling limit and extend critical-curve regularity and sharpness to a class of long-range frog models. The main tool is a local-to-global principle for activation processes generated by independent finite rooted ranges: local one-hit comparison implies comparison of global reachability and survival.

math.PR↗

arXiv2026

Free-Energy Asymptotics of the Two-Dimensional Quantum Heisenberg Ferromagnet

We prove the leading low-temperature free-energy asymptotics of the nearest-neighbour quantum Heisenberg ferromagnet on $\mathbb Z^2$. For every fixed $S\in\{\frac12,1,\frac32,\ldots\}$, the free energy per site satisfies \[ \lim_{β\to\infty}β^2S f_2(β,S)=-\fracπ{24}. \] This formula was predicted by Takahashi's modified spin-wave theory [15]. Napiórkowski and Seiringer proved the corresponding upper bound but left the matching lower bound in two dimensions open [13], a gap later highlighted again by Seiringer [14]. We prove this missing bound through a new comparison between the magnon exclusion dynamics and free bosons, based on extending wave functions to collision configurations with controlled kinetic cost.

math-ph↗

arXiv2026

(Non-)coincidence of critical parameters for Poisson Zoos and Loop Soup Percolation on $\mathbb{Z^d}$, $d > 4$, and $\mathbb{T}_d$,$ d \ge 3$

In this article we investigate the (non)-coincidence of critical parameters for various related percolation problems. More precisely, for the random walk loop soup we show that on $\mathbb Z^d$, $d\ge 5$, the critical parameters for the percolation problems differ on the discrete graph and the respective metric graph. Moreover, on trees we deduce an analogous statement as well as the coincidence of the critical parameters for percolation and susceptibility for a more general class of percolation problems, the so-called Poisson zoo. Along the way we develop the useful notion of sensitivity to Bernoulli enhancements of such percolation problems with long range correlations, which builds on previously developed enhancement ideas.

math.PR↗

arXiv2026

Loop vs. Bernoulli percolation on trees: strict inequality of critical values

We study loop ensembles on locally finite rooted trees. The loops are induced by a Poisson process of links on the edges; the transposition-only case is the random interchange process, and the same framework covers loop representations of quantum spin systems. The set of edges carrying at least one untyped link forms i.i.d. Bernoulli bond percolation with retention probability $1 - \exp(-β)$, so an infinite loop can occur only if the corresponding link cluster is infinite. Our main results for Galton-Watson trees show both phenomena: if the offspring distribution has finite mean m in $(1,\infty)$, then, conditioned on survival, the loop threshold is strictly larger than the link threshold; in contrast, in the random interchange case a heavy-tail condition on the offspring distribution forces the loop and link thresholds, averaged over both the tree and the link configuration, to coincide at zero. The separation theorem follows from a stronger deterministic criterion based on local loop configurations that cut descendant subtrees out of link clusters.

math.PR↗

arXiv2026

Macroscopic loops in the random loop model on sparse random graphs

We study the random loop model with crosses and bars on sparse random graphs. Our main objective is to prove the existence of macroscopic loops, in the sense that a loop visits a positive proportion of the vertices. We develop a deterministic drift method on arbitrary finite graphs based on three ingredients: a local split--merge--rewire analysis for the loop number, an exact differential identity for the partition function, and a slice estimate reducing the relevant same-loop insertion volume to induced edge counts of small vertex sets. This yields a general criterion in terms of a small-set sparsity condition on the underlying graph. We then verify this condition for random regular graphs, sparse Erdős--Rényi graphs, and simple bounded-degree configuration models, obtaining averaged lower bounds on the probability of a macroscopic loop whenever the edge density exceeds an explicit threshold depending on the loop weight $θ$ and the cross parameter $u$. For integer values of $θ$, a trace representation of the partition function implies log-convexity, which upgrades the averaged bounds to pointwise-in-time results away from the threshold time.

math.PR↗

arXiv2025

Extremes of the zero-average Gaussian Free Field on random regular graphs

We study the extreme value statistics of the zero-average Gaussian free field (GFF) on random $r$-regular graphs and the Gaussian free field on $r$-regular trees. For random $r$-regular graphs of diverging size, for every fixed $r\ge3$, we show that the rescaled extremal point process of the field is asymptotically distributed, in the annealed sense, as a Poisson point process on the line with intensity $e^{-x}\,\mathrm{d}x$. The same limit behaviour is obeyed by the restriction of the GFF on $r$-regular trees to finite subsets of vertices. Our approach relies on a direct Gaussian comparison argument and precise Green function estimates.

math.PR↗

arXiv2025

Loop percolation versus link percolation in the random loop model

In [Muhl2019], Peter Mühlbacher showed that in the random loop model without loop weights, a loop phase transition (assuming it exists) cannot occur at the same parameter as the percolation phase transition of the occupied edges. In this work, we give a quantitative version of this result, specifying a minimal gap between the percolation phase transition and a possible loop phase transition. A substantial part of our argument also works for weighted loop models.

math.PR↗

arXiv2025

Improved bounds for connection probabilities in random loop models

We revisit and extend results by Ueltschi [19] on the application of reflection positivity to loop models with $θ\in \mathbb{N}_{\geq 2}$. By exploiting additional flexibility in the method, we prove the existence of long loops over a broader range of parameters $u$ and $θ$, and establish new lower bounds for connection probabilities and the critical parameter $β_c$. Our results are compared with recent numerical simulations, providing further insight into the phase diagram of quantum spin systems.

math-ph↗