arXiv subjects

Diederik van Engelenburg

Publications and source records attributed to Diederik van Engelenburg.

9 recordsLinked to original sources

arXiv2026

Double cluster swapping for spin models: Pfaffian relations and sharpness

We introduce and study a new geometric representation for general classical spin models. It consists of two coupled percolation configurations that are the joint FK (random cluster) representation of the Ginibre rotation of two independent copies of the spin model, and can be viewed as an extension of Sheffield's cluster swapping defined in the context of height functions. Our approach combines the advantages of the random current and FK representations of the Ising model: it provides a percolation interpretation for various truncated correlation functions and at the same time satisfies the FKG inequality (for a large subclass of models). We highlight its strength and versatility by establishing two very different results. We first prove the converse of the classical fact that boundary multi-point correlation functions of planar Ising models are given by Pfaffians of the two-point functions. Indeed, we show that if a general spin model on a general graph satisfies the Pfaffian relations, then up to natural local modifications it must actually be an Ising model on a planar graph. In particular, the algebraic Pfaffian relations imply the topological feature of planarity. Our second application is a proof of sharpness of the phase transition for a class of spin models first considered by Ellis, Monroe and Newman, which we do by generalising the celebrated argument of Duminil-Copin and Tassion, replacing the use of the random current with our new representation.

math.PR↗

arXiv2026

The impact of disorder and non-convex interactions on delocalisation of height functions

We study the behaviour of four spins systems (the XY model, the Villain model, the XY height function and the integer-valued Gaussian free field) in the presence of a non-elliptic quenched disorder. In the article [DG25], it was shown that the phase transitions of the XY model (the Berezinskii-Kosterlitz-Thouless phase transition in $d = 2$ and the order/disorder phase transition when $d \geq 3$) persist on the infinite cluster of a supercritical Bernoulli percolation. A first objective of this article is to extend these results to the Villain model. Our second objective is to analyse, for $d=2$, how the corresponding dual integer-valued height function models behave in the presence of a dual quenched disorder. These dual models are respectively the XY height function and the integer-valued Gaussian free field. Without disorder, these models are known to exhibit a phase transition in two dimensions called the roughening transition [FS81, Lam22b]. We show that this phase transition persists when the quenched disorder is given by enforcing $φ(x) = φ(y)$ independently with probability $\bar{p} < 1/2$ for neighboring sites $x, y$. Finally, we apply our methods to integer-valued height functions with annealed Gaussian interactions and prove the existence of a (quantified) rough phase. This includes all potentials of the form $|\nabla h|^p$ for $p \in (0, 2]$, recovering recent results of [OS25].

math.PR↗

arXiv2025

One-arm exponents of the high-dimensional Ising model

We study the probability that the origin is connected to the boundary of the box of size $n$ (the one-arm probability) in several percolation models related to the Ising model. We prove that different universality classes emerge at criticality. - For the FK-Ising measure in a box of size $n$ with wired boundary conditions, we prove that this probability decays as $1/n$ in dimensions $d>4$, and as $1/n^{1+o(1)}$ when $d=4$. - For the infinite volume FK-Ising measure, we prove that this probability decays as $1/n^2$ in dimensions $d>6$, and as $1/n^{2+o(1)}$ when $d=6$. - For the sourceless double random current measure, we prove that this probability decays as $1/n^{d-2}$ in dimensions $d>4$, and as $1/n^{2+o(1)}$ when $d=4$. Additionally, for the infinite volume FK-Ising measure, we show that the one-arm probability is $1/n^{1+o(1)}$ in dimension $d=4$, and at least $1/n^{3/2}$ in dimension $d=5$. This establishes that the FK-Ising model has upper-critical dimension equal to $6$, in contrast to the Ising model, where it is known to be less or equal to $4$, thus solving a conjecture of Chayes, Coniglio, Machta, and Shtengel.

math.PR↗

arXiv2025

One-arm exponents of high-dimensional percolation revisited

We consider sufficiently spread-out Bernoulli percolation in dimensions ${d>6}$. We present a short and simple proof of the up-to-constants estimate for the one-arm probability in both the full-space and half-space settings. These results were previously established by Kozma and Nachmias and by Chatterjee and Hanson, respectively. Our proof improves upon the entropic technique introduced by Dewan and Muirhead, relying on a sharp estimate on a suitably chosen correlation length recently obtained by Duminil-Copin and Panis. This approach is inspired by our companion work, where we compute the one-arm exponent for several percolation models related to the high-dimensional Ising model.

math.PR↗

arXiv2023

On the duality between height functions and continuous spin models

We revisit the classical phenomenon of duality between random integer-valued height functions with positive definite potentials and abelian spin models with O(2) symmetry. We use it to derive new results in quite high generality including: a universal upper bound on the variance of the height function in terms of the Green's function (a GFF bound) which among others implies localisation on transient graphs; monotonicity of said variance with respect to a natural temperature parameter; the fact that delocalisation of the height function implies a BKT phase transition in planar models; and also delocalisation itself for height functions on periodic ``almost'' planar graphs.

math.PR↗

arXiv2023

The number of ends in the uniform spanning tree for recurrent unimodular random graphs

We prove that if a unimodular random rooted graph is recurrent, the number of ends of its uniform spanning tree is almost surely equal to the number of ends of the graph. Together with previous results in the transient case, this completely resolves the problem of the number of ends of wired uniform spanning forest components in unimodular random rooted graphs and confirms a conjecture of Aldous and Lyons (2006).

math.PR↗

arXiv2021

An elementary proof of phase transition in the planar XY model

Using elementary methods we obtain a power-law lower bound on the two-point function of the planar XY spin model at low temperatures. This was famously first rigorously obtained by Fröhlich and Spencer and establishes a Berezinskii-Kosterlitz-Thouless phase transition in the model. Our argument relies on a new loop representation of spin correlations, a recent result of Lammers on delocalisation of integer-valued height functions, and classical correlation inequalities.

math-ph↗

arXiv2021

Harnack inequality and one-endedness of UST on reversible random graphs

We prove that for recurrent, reversible graphs, the following conditions are equivalent: (a) existence and uniqueness of the potential kernel, (b) existence and uniqueness of harmonic measure from infinity, (c) a new anchored Harnack inequality, and (d) one-endedness of the wired Uniform Spanning Tree. In particular this gives a proof of the anchored (and in fact also elliptic) Harnack inequality on the UIPT. This also complements and strengthens some results of Benjamini, Lyons, Peres and Schramm. Furthermore, we make progress towards a conjecture of Aldous and Lyons by proving that these conditions are fulfilled for strictly subdiffusive recurrent unimodular graphs. Finally, we discuss the behaviour of the random walk conditioned to never return to the origin, which is well defined as a consequence of our results

math.PR↗

arXiv2020

An upper bound on the two-arms exponent for critical percolation on $\mathbb{Z}^d$

Consider critical site percolation on $\mathbb{Z}^d$ with $d \geq 2$. Cerf (2015) pointed out that from classical work by Aizenman, Kesten and Newman (1987) and Gandolfi, Grimmett and Russo (1988) one can obtain that the two-arms exponent is at least $1/2$. The paper by Cerf slightly improves that lower bound. Except for $d=2$ and for high $d$, no upper bound for this exponent seems to be known in the literature so far (not even implicity). We show that the distance-$n$ two-arms probability is at least $c n^{-(d^2 + 4 d -2)}$ (with $c >0$ a constant which depends on $d$), thus giving an upper bound $d^2 + 4 d -2$ for the above mentioned exponent.

math.PR↗