arXiv subjects

Marcin Lis

Publications and source records attributed to Marcin Lis.

At least 19 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 Ising magnetisation field and the Gaussian free field

We construct a natural coupling between the continuum Gaussian free field (GFF) and the critical Ising magnetisation field (IMF) in a planar domain. In fact, we show that two independent IMFs with $+$ boundary conditions and two independent IMFs with free boundary conditions are a deterministic function of a single instance of the GFF together with a sequence of independent coin flips. This construction should be seen as an extension of the bosonisation phenomenon, and to the best of our knowledge its existence has not been predicted before. We arrive at our main result in the continuum by studying novel discrete structures. Our starting point is a coupling resembling the Edwards-Sokal coupling between the Ising model and the Fortuin-Kasteleyn random cluster model, though with role of the latter played by a different percolation model obtained from the double random current model. By taking a scaling limit of the coupling at criticality, we obtain a continuum Edwards-Sokal-like representation of the IMFs in terms of certain two-valued sets of the GFF introduced by Aru, Sep\'ulveda and Werner.

math.PR↗

arXiv2025

Uniform spanning trees and random matrix statistics

We consider a uniform spanning tree in a $\delta$-square grid approximation of a planar domain $\Omega$. For given integer $n\ge 2$, we condition the tree on the following $n$-arm event: we pick $n$ branches, emanating from $n$ points microscopically close to a given interior point, and condition them to connect to the boundary $\partial \Omega$ without intersecting. What can be said about the geometry of these branches? We derive an exact formula for the characteristic function of the total winding of the branches. A surprising consequence of this formula is that in the scaling limit, the behaviour of this function depends on the total number of branches $n$ only through its parity. We also describe the scaling limit of the branches. If $\Omega $ is the unit disc, then they hit the boundary (i.e., the unit circle) at random positions which coincide exactly with the eigenvalues of a random matrix of size $n$ drawn from the Circular Orthogonal Ensemble (COE, also called C$\beta$E with $\beta =1$). Furthermore, the branches converge to Loewner evolution driven by the circular Dyson Brownian motion with parameter $\beta = 4$ (i.e., $n$-sided radial SLE$_\kappa$ with $\kappa=2$). We thus verify a prediction made by Cardy in this setting. Along the way, we develop a flow-line (imaginary geometry) coupling of $n$-sided radial SLE$_\kappa$ with the Gaussian free field, which may be of independent interest. Surprisingly, we find that the variance of the corresponding field near the singularity also does not depend on the number $n\ge 2$ of curves. In contrast, the variance of the the winding of the curves behaves as $\kappa/n^2$, which agrees with the predictions from the physics literature made by Wieland and Wilson numerically, and by Duplantier and Binder using Coulomb gas methods -- but disagrees with a result of Kenyon.

math.PR↗

arXiv2024

Zeros of planar Ising models via flat SU(2) connections

Livine and Bonzom recently proposed a geometric formula for a certain set of complex zeros of the partition function of the Ising model defined on planar graphs. Remarkably, the zeros depend locally on the geometry of an immersion of the graph in the three dimensional Euclidean space (different immersions give rise to different zeros). When restricted to the flat case, the weights become the critical weights on circle patterns. We rigorously prove the formula by geometrically constructing a null eigenvector of the Kac-Ward matrix whose determinant is the squared partition function. The main ingredient of the proof is the realisation that the associated Kac-Ward transition matrix gives rise to an SU(2) connection on the graph, creating a direct link with rotations in three dimensions. The existence of a null eigenvector turns out to be equivalent to this connection being flat.

math-ph↗

arXiv2024

Conformally invariant boundary arcs in double dimers

We consider two different versions of the double dimer model on a planar domain, where we either fold a single dimer cover on a symmetric domain onto itself across the line of symmetry, or we superimpose two independent dimer covers on two, almost identical, domains that differ only on a certain portion of the boundary. This results in a collection of loops and doubled edges that, unlike in the classical double dimer case of Kenyon, are accompanied by arcs emanating from the line of symmetry or the chosen portion of the boundary. We argue that these arcs together with the associated height function satisfy a discrete version of the coupling of Qian and Werner between the Arc loop ensemble (ALE) and two different variants of the Gaussian free field (with Dirichlet and Neumann boundary conditions). We also show that certain statistics of the arcs (when the loops are disregarded from the picture) converge to conformally invariant quantities in the small-mesh scaling limit, and moreover the limits are the same for the two versions of the model, and equal to the corresponding statistics of the arc loop ensemble (ALE). This gives evidence to the conjecture of [7] (that concerns one of these models).

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↗

arXiv2021

Conformal invariance of double random currents II: tightness and properties in the discrete

This is the second of two papers devoted to the proof of conformal invariance of the critical double random current on the square lattice. More precisely, we show convergence of loop ensembles obtained by taking the cluster boundaries in the sum of two independent critical currents (both for free and wired boundary conditions). The strategy is first to prove convergence of the associated height function to the continuum Gaussian free field, and then to characterize the scaling limit of the loop ensembles as certain local sets of this Gaussian Free Field. In this paper, we derive crossing properties of the discrete model required to prove this characterization.

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

Conformal invariance of double random currents I: identification of the limit

This is the first of two papers devoted to the proof of conformal invariance of the critical double random current model on the square lattice. More precisely, we show the convergence of loop ensembles obtained by taking the cluster boundaries in the sum of two independent currents with free and wired boundary conditions. The strategy is first to prove convergence of the associated height function to the continuum Gaussian free field, and then to characterize the scaling limit of the loop ensembles as certain local sets of this Gaussian Free Field. In this paper, we identify uniquely the possible subsequential limits of the loop ensembles. Combined with the second paper, this completes the proof of conformal invariance.

math.PR↗

arXiv2021

Free boundary dimers: random walk representation and scaling limit

We study the dimer model on subgraphs of the square lattice in which vertices on a prescribed part of the boundary (the free boundary) are possibly unmatched. Each such unmatched vertex is called a monomer and contributes a fixed multiplicative weight $z>0$ to the total weight of the configuration. A bijection described by Giuliani, Jauslin and Lieb relates this model to a standard dimer model but on a non-bipartite graph. The Kasteleyn matrix of this dimer model describes a walk with transition weights that are negative along the free boundary. Yet under certain assumptions, which are in particular satisfied in the infinite volume limit in the upper half-plane, we prove an effective, true random walk representation for the inverse Kasteleyn matrix. In this case we further show that, independently of the value of $z>0$, the scaling limit of the height function is the Gaussian free field with Neumann (or free) boundary conditions, thereby answering a question of Giuliani et al.

math.PR↗

arXiv2021

On delocalization in the six-vertex model

We show that the six-vertex model with parameter $c\in[\sqrt 3, 2]$ on a square lattice torus has an ergodic infinite-volume limit as the size of the torus grows to infinity. Moreover we prove that for $c\in[\sqrt{2+\sqrt 2}, 2]$, the associated height function on $\mathbb Z^2$ has unbounded variance. The proof relies on an extension of the Baxter-Kelland-Wu representation of the six-vertex model to multi-point correlation functions of the associated spin model. Other crucial ingredients are the uniqueness and percolation properties of the critical random cluster measure for $q\in[1,4]$, and recent results relating the decay of correlations in the spin model with the delocalization of the height function.

math.PR↗

arXiv2020

On boundary correlations in planar Ashkin-Teller models

We generalize the switching lemma of Griffiths, Hurst and Sherman to the random current representation of the Ashkin-Teller model. We then use it together with properties of two-dimensional topology to derive linear relations for multi-point boundary spin correlations and bulk order-disorder correlations in planar models. We also show that the same linear relations are satisfied by products of Pfaffians. As a result a clear picture arises in the noninteracting case of two independent Ising models where multi-point correlation functions are given by Pfaffians and determinants of their respective two-point functions. This gives a unified treatment of both the classical Pfaffian identities and recent total positivity inequalities for boundary spin correlations in the planar Ising model. We also derive the Simon and Gaussian inequality for general Ashkin-Teller models with negative four-body coupling constants.

math-ph↗

arXiv2020

Spins, percolation and height functions

To highlight certain similarities in combinatorial representations of several well known two-dimensional models of statistical mechanics, we introduce and study a new family of models which specializes to these cases after a proper tuning of the parameters. To be precise, our model consists of two independent standard Potts models, with possibly different numbers of spins and different coupling constants (the four parameters of the model), defined jointly on a graph embedded in a surface and its dual graph, and conditioned on the event that the primal and dual interfaces between spins of different value do not intersect. We also introduce naturally related height function and bond percolation models, and we discuss their basic properties and mutual relationship. As special cases we recover the standard Potts and random cluster model, the six-vertex model and loop $O(n)$ model, the random current, double random current and XOR-Ising model.

math.PR↗

arXiv2019

Circle patterns and critical Ising models

A circle pattern is an embedding of a planar graph in which each face is inscribed in a circle. We define and prove magnetic criticality of a new family of Ising models on planar graphs whose dual is a circle pattern. Our construction includes as a special case the critical isoradial Ising models of Baxter.

math-ph↗

arXiv2019

On the double random current nesting field

We relate the planar random current representation introduced by Griffiths, Hurst and Sherman to the dimer model. More precisely, we provide a measure-preserving map between double random currents (obtained as the sum of two independent random currents) on a planar graph and dimers on an associated bipartite graph. We also construct a nesting field for the double random current, which, under this map, corresponds to the height function of the dimer model. As applications, we provide an alternative derivation of some of the bozonization rules obtained recently by Dubédat, and show that the spontaneous magnetization of the Ising model on a planar biperiodic graph vanishes at criticality.

math-ph↗

arXiv2018

Spin systems from loop soups

We study spin systems defined by the winding of a random walk loop soup. For a particular choice of loop soup intensity, we show that the corresponding spin system is reflection-positive and is dual, in the Kramers-Wannier sense, to the spin system $\text{sgn}(φ)$ where $φ$ is a discrete Gaussian free field. In general, we show that the spin correlation functions have conformally covariant scaling limits corresponding to the one-parameter family of functions studied by Camia, Gandolfi and Kleban (Nuclear Physics B, 902, 2016) and defined in terms of the winding of the Brownian loop soup. These functions have properties consistent with the behavior of correlation functions of conformal primaries in a conformal field theory. Here, we prove that they do correspond to correlation functions of continuum fields (random generalized functions) for values of the intensity of the Brownian loop soup that are not too large.

math-ph↗

arXiv2016

The planar Ising model and total positivity

A matrix is called totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative). Consider the Ising model with free boundary conditions and no external field on a planar graph $G$. Let $a_1,\dots,a_k,b_k,\dots,b_1$ be vertices placed in a counterclockwise order on the outer face of $G$. We show that the $k\times k$ matrix of the two-point spin correlation functions \[ M_{i,j} = \langle σ_{a_i} σ_{b_j} \rangle \] is totally nonnegative. Moreover, $\det M > 0$ if and only if there exist $k$ pairwise vertex-disjoint paths that connect $a_i$ with $b_i$. We also compute the scaling limit at criticality of the probability that there are $k$ parallel and disjoint connections between $a_i$ and $b_i$ in the double random current model. Our results are based on a new distributional relation between double random currents and random alternating flows of Talaska.

math.PR↗

arXiv2016

Non-backtracking loop soups and statistical mechanics on spin networks

We introduce and study a Markov field on the edges of a graph in dimension $d\geq2$ whose configurations are spin networks. The field arises naturally as the edge-occupation field of a Poissonian model (a soup) of non-backtracking loops and walks characterized by a spatial Markov property such that, conditionally on the value of the edge-occupation field on a boundary set, the distributions of the loops and arcs on either side of the boundary are independent of each other. The field has a Gibbs distribution with a Hamiltonian given by a sum of terms which involve only edges incident on the same vertex. Its free energy density and other quantities can be computed exactly, and their critical behavior analyzed, in any dimension.

math-ph↗