arXiv subjects

Piet Lammers

Publications and source records attributed to Piet Lammers.

15 recordsLinked to original sources

arXiv2026

The BKT transition and surface tension differentiability

Under the duality between the two-dimensional XY model and an integer-valued height function, the BKT transition is expected to correspond to the disappearance of a corner in the surface tension; in the delocalised phase, its zero-slope curvature should determine the Gaussian free field prefactor. We prove that the XY mass equals the right derivative at zero slope of the dual height function's free energy, with a uniform quadratic error bound near the origin. Thus the massive phase is exactly the corner regime, while in the BKT phase the surface tension is quadratically bounded at zero slope. The proof combines Kadanoff--Ceva duality, Ginibre's inequality, and a pushing lemma (an estimate of Russo--Seymour--Welsh type) for the cable height function.

math.PR↗

arXiv2026

Free energy analyticity of the disordered XY model and Debye screening in the 2D Coulomb gas

We consider three models of statistical mechanics: the classical XY model in arbitrary dimension, the lattice Coulomb gas in dimension two, and the square well model in arbitrary dimension. For each of these three models, we prove that the free energy is analytic in the disordered regime (the square well model is disordered at any positive temperature). In order to prove these results, we prove that the Gibbs measures of these models are factors of i.i.d. with information clusters of exponentially decaying size (volume). In the case of the Coulomb gas, we obtain a strong version of Debye screening with an arbitrary number of arbitrary local observables of the Coulomb gas, and we prove that the Debye phase contains the complement of the Berezinskii-Kosterlitz-Thouless phase.

math.PR↗

arXiv2026

Gaussian free field convergence of the six-vertex model with $-1\leq\Delta\leq-\frac12$

We study the isotropic six-vertex model on $\mathbb{Z}^2$ with spectral parameter $\Delta\in[-1,-1/2]$, that is, with weights $\mathbf{a}=\mathbf{b}=1$ and $\mathbf{c}\in[\sqrt{3},2]$. We show that the associated height function converges, in the scaling limit, to a properly scaled full-plane Gaussian free field. The result extends to anisotropic weights $\mathbf{a}\neq\mathbf{b}$ upon using a suitable embedding of the lattice.

math-ph↗

arXiv2023

Delocalisation and continuity in 2D: loop O(2), six-vertex, and random-cluster models

We prove the existence of macroscopic loops in the loop O(2) model with $\frac12\leq x^2\leq 1$ or, equivalently, delocalisation of the associated integer-valued Lipschitz function on the triangular lattice. This settles one side of the conjecture of Fan, Domany, and Nienhuis (1970s-80s) that $x^2 = \frac12$ is the critical point. We also prove delocalisation in the six-vertex model with $0<a,\,b\leq c\leq a+b$. This yields a new proof of continuity of the phase transition in the random-cluster and Potts models in two dimensions for $1\leq q\leq 4$ relying neither on integrability tools (parafermionic observables, Bethe Ansatz), nor on the Russo-Seymour-Welsh theory. Our approach goes through a novel FKG property required for the non-coexistence theorem of Zhang and Sheffield, which is used to prove delocalisation all the way up to the critical point. We also use the $\mathbb T$-circuit argument in the case of the six-vertex model. Finally, we extend an existing renormalisation inequality in order to quantify the delocalisation as being logarithmic, in the regimes $\frac12\leq x^2\leq 1$ and $a=b\leq c\leq a+b$. This is consistent with the conjecture that the scaling limit is the Gaussian free field.

math.PR↗

arXiv2023

Bijecting the BKT transition

We consider the classical XY model (or classical rotor model) on the two-dimensional square lattice graph as well as its dual model, which is a model of height functions. The XY model has a phase transition called the Berezinskii-Kosterlitz-Thouless transition. There is a heuristic which predicts that this phase transition should coincide with the localisation-delocalisation transition for height functions: the primal and dual model share the same partition function, and the phase transition of either model should coincide with the unique non-analytic point of the partition function when expressed in terms of the inverse temperature. We use probabilistic arguments to prove that the correlation length (the reciprocal of the mass) of the XY model is exactly twice the correlation length of the height function, which implies in particular that the prediction of this heuristic is correct: namely, that the BKT phase for the XY model coincides exactly with the delocalised phase of the dual height function.

math.PR↗

arXiv2022

A dichotomy theory for the height functions of the BKT transition

This text considers the discrete height functions associated with the Berezinskii--Kosterlitz--Thouless transition (BKT) at slope zero. Our main results are as follows. * Sharpness: If the model is localised, then the two-point function (covariance) decays exponentially fast in the distance between the points. * Effective temperature gap: If the model is delocalised, then the variance grows at least as $c\log n$, where $n$ is the distance to the boundary and $c>0$ a universal constant not depending on the temperature. Thus, the effective temperature must jump from $0$ to at least $c$ at the transition point; values in the interval $(0,c)$ are forbidden. * Delocalisation at the transition point: The delocalised phase includes the transition point, in the sense that it is a closed set in the phase diagram in the appropriate topology. These results contribute to the understanding of the regime at and around the transition point which remained largely unexplored. In a follow-up paper, the sharpness derived here is used to establish that the localisation-delocalisation transition is equivalent to the BKT transition in the dual XY and Villain models.

math.PR↗

arXiv2022

Non-reversible stationary states for majority voter and Ising dynamics on trees

We study three Markov processes on infinite, unrooted, regular trees: the stochastic Ising model (also known as the Glauber heat bath dynamics of the Ising model), a majority voter dynamic, and a coalescing particle model. In each of the three cases the tree exhibits a preferred direction encoded into the model. For all three models, our main result is the existence of a stationary but non-reversible measure. For the Ising model, this requires imposing that the inverse temperature is large and choosing suitable non-uniform couplings, and our theorem implies the existence of a stationary measure which looks nothing like a low-temperature Gibbs measure. The interesting aspect of our results lies in the fact that the analogous processes do not have non-Gibbsian stationary measures on $\mathbb Z^d$, owing to the amenability of that graph. In fact, no example of a stochastic Ising model with a non-reversible stationary state was known to date.

math.PR↗

arXiv2022

Height function localisation on trees

We study two models of discrete height functions, that is, models of random integer-valued functions on the vertices of a tree. First, we consider the random homomorphism model, in which neighbours must have a height difference of exactly one. The local law is uniform by definition. We prove that the height variance of this model is bounded, uniformly over all boundary conditions (both in terms of location and boundary heights). This implies a strong notion of localisation, uniformly over all extremal Gibbs measures of the system. For the second model, we consider directed trees, in which each vertex has exactly one parent and at least two children. We consider the locally uniform law on height functions which are monotone, that is, such that the height of the parent vertex is always at least the height of the child vertex. We provide a complete classification of all extremal gradient Gibbs measures, and describe exactly the localisation-delocalisation transition for this model. Typical extremal gradient Gibbs measures are localised also in this case. Localisation in both models is consistent with the observation that the Gaussian free field is localised on trees, which is an immediate consequence of transience of the random walk.

math.PR↗

arXiv2021

Height function delocalisation on cubic planar graphs

The interest is in models of integer-valued height functions on shift-invariant planar graphs whose maximum degree is three. We prove delocalisation for models induced by convex nearest-neighbour potentials, under the condition that each potential function is an excited potential, that is, a convex symmetric potential function $V$ with the property that $V(\pm1)\leq V(0)+\log2$. Examples of such models include the discrete Gaussian and solid-on-solid models at inverse temperature $β\leq\log2$, as well as the uniformly random $K$-Lipschitz function for fixed $K\in\mathbb N$. In fact, $βV$ is an excited potential for any convex symmetric potential function $V$ whenever $β$ is sufficiently small. To arrive at the result, we develop a new technique for symmetry breaking, and then study the geometric percolation properties of sets of the form $\{φ\geq a\}$ and $\{φ\leq a\}$, where $φ$ is the random height function and $a$ a constant. Along the same lines, we derive delocalisation for models induced by convex symmetric nearest-neighbour potentials which force the parity of the height of neighbouring vertices to be distinct. This includes models of uniformly random graph homomorphisms on the honeycomb lattice and the truncated square tiling, as well as on the same graphs with each edge replaced by $N$ edges linked in series. The latter resembles cable-graph constructions which appear in the analysis of the Gaussian free field.

math.PR↗

arXiv2021

Variational principle for weakly dependent random fields

Using an alternative notion of entropy introduced by Datta, the max-entropy, we present a new simplified framework to study the minimizers of the specific free energy for random fields which are weakly dependent in the sense of Lewis, Pfister, and Sullivan. The framework is then applied to derive the variational principle for the loop $O(n)$ model and the Ising model in a random percolation environment in the nonmagnetic phase, and we explain how to extend the variational principle to similar models. To demonstrate the generality of the framework, we indicate how to naturally fit into it the variational principle for models with an absolutely summable interaction potential, and for the random-cluster model.

math.PR↗

arXiv2021

The bunkbed conjecture on the complete graph

The bunkbed conjecture was first posed by Kasteleyn. If $G=(V,E)$ is a finite graph and $H$ some subset of $V$, then the bunkbed of the pair $(G,H)$ is the graph $G\times\{1,2\}$ plus $|H|$ extra edges to connect for every $v\in H$ the vertices $(v,1)$ and $(v,2)$. The conjecture asserts that $(v,1)$ is more likely to connect with $(w,1)$ than with $(w,2)$ in the independent bond percolation model for any $v,w\in V$. This is intuitive because $(v,1)$ is in some sense closer to $(w,1)$ than it is to $(w,2)$. The conjecture has however resisted several attempts of proof. This paper settles the conjecture in the case of a constant percolation parameter and $G$ the complete graph.

math.CO↗

arXiv2021

A generalisation of the honeycomb dimer model to higher dimensions

Linde, Moore, and Nordahl introduced a generalisation of the honeycomb dimer model to higher dimensions. The purpose of this article is to describe a number of structural properties of this generalised model. First, it is shown that the samples of the model are in one-to-one correspondence with the perfect matchings of a hypergraph. This leads to a generalised Kasteleyn theory: the partition function of the model equals the Cayley hyperdeterminant of the adjacency hypermatrix of the hypergraph. Second, we prove an identity which relates the covariance matrix of the random height function directly to the random geometrical structure of the model. This identity is known in the planar case but is new for higher dimensions. It relies on a more explicit formulation of Sheffield's cluster swap which is made possible by the structure of the honeycomb dimer model. Finally, we use the special properties of this explicit cluster swap to give a new and simplified proof of strict convexity of the surface tension in this case.

math.PR↗

arXiv2021

Delocalisation and absolute-value-FKG in the solid-on-solid model

The solid-on-solid model is a model of height functions, introduced to study the interface separating the $+$ and $-$ phase in the Ising model. The planar solid-on-solid model thus corresponds to the three-dimensional Ising model. Delocalisation of this model at high temperature and at zero slope was first derived by Fr\"ohlich and Spencer, in parallel to proving the Berezinskii-Kosterlitz-Thouless phase transition. The first main result of this article consists of a simple, alternative proof of delocalisation of the solid-on-solid model. In fact, the argument is more general: it works on any planar graph -- not just the square lattice -- and implies that the interface delocalises at any slope rather than exclusively at the zero slope. The second main result, proved independently, is that the absolute value of the height function in this model satisfies the FKG lattice condition. This property is believed to be intimately linked to the (quantitative) understanding of delocalisation, given the recent successes in the context of the square ice and (more generally) the six-vertex model, and it has already been used elsewhere in a new proof of the BKT transition. The new FKG inequality is shown to hold true for both the solid-on-solid model as well as for the discrete Gaussian model, which in this article implies that the two notions of delocalisation, namely delocalisation in finite volume and delocalisation of shift-invariant Gibbs measures, coincide.

math.PR↗

arXiv2020

Macroscopic behavior of Lipschitz random surfaces

The motivation for this article is to derive strict convexity of the surface tension for Lipschitz random surfaces, that is, for models of random Lipschitz functions from $\mathbb Z^d$ to $\mathbb Z$ or $\mathbb R$. An essential innovation is that random surface models with long- and infinite-range interactions are included in the analysis. More specifically, we cover at least: uniformly random graph homomorphisms from $\mathbb Z^d$ to a $k$-regular tree for any $k\geq 2$ and Lipschitz potentials which satisfy the FKG lattice condition. The latter includes perturbations of dimer- and six-vertex models and of Lipschitz simply attractive potentials introduced by Sheffield. The main result is that we prove strict convexity of the surface tension -- which implies uniqueness for the limiting macroscopic profile -- if the model of interest is monotone in the boundary conditions. This solves a conjecture of Menz and Tassy, and answers a question posed by Sheffield. Auxiliary to this, we prove several results which may be of independent interest, and which do not rely on the model being monotone. This includes existence and topological properties of the specific free energy, as well as a characterization of its minimizers. We also prove a general large deviations principle which describes both the macroscopic profile and the local statistics of the height functions. This work is inspired by, but independent of, Random Surfaces by Sheffield.

math.PR↗

arXiv2017

Diffusivity of a walk on fractures of a hypertorus

This article studies discrete height functions on the discrete hypertorus. These are functions on the vertices of this hypertorus graph for which the derivative satisfies a specific condition on each edge. We then perform a random walk on the set of such height functions, in the spirit of Diffusivity of a random walk on random walks, a work of Boissard, Cohen, Espinasse, and Norris. The goal is to estimate the diffusivity of this random walk in the mesh limit. It turns out that each height functions is characterised by a number of so-called fractures of the hypertorus. These fractures are then studied in isolation; we are able to understand their asymptotic behaviour in the mesh limit due to the recent understanding of the associated random surfaces. This allows for an asymptotic reduction to a one-dimensional continuous system consisting of $\operatorname{gcd}\mathbf n$ parts where $\mathbf n\in\mathbb N^d$ is the fundamental parameter of the original model. We then prove that the diffusivity of the random walk tends to $1/(1+2\operatorname{gcd}\mathbf n)$ in this mesh limit.

math.PR↗