Alexander Glazman
University of InnsbruckAss. Prof. Alexander Glazman, PhD
Phone: +43 512 507 53930
About me
Loop O(n) model, Sixvertex model, AshkinTeller model, randomcluster model, SelfAvoiding Walk, Ising model.
Teaching
 Winter 2023/24: Introduction to Higher Stochastics (Einführung in die Höhere Stochastik)
 Summer 2023: Discrete Probability in models of Mathematical Physics, more information
 Winter 2022/23: Introduction to Higher Stochastics (Einführung in die Höhere Stochastik)
Papers

Delocalisation and Continuity in 2D: Loop \(O(2)\), SixVertex, and RandomCluster Models, Preprint, with Piet Lammers
We prove the existence of macroscopic loops in the loop \(O(2)\) model with \(\frac12\leq x^2\leq 1\). This implies a logarithmic delocalisation of an integervalued Lipschitz function on the triangular lattice and settles one side of the conjecture of Fan, Domany, and Nienhuis: they predicted in the 1970s80s that \(x^2=\frac12\) is the critical point for the localisationdelocalisation transition.
We also prove delocalisation in the sixvertex model with \(0<a,b\leq c\leq a+b\). This yields a new proof of continuity of the phase transition in the randomcluster and Potts models in two dimensions, for \(1\leq q\leq 4\). We rely neither on integrability tools (parafermionic observables, Bethe Ansatz), nor on the RussoSeymourWelsh theory.
Our approach goes through a novel FKG property that enables the use of the noncoexistence theorem of Zhang and Sheffield to prove delocalisation all the way up to the critical point. In the sixvertex model, we additionally use the \(\mathbb T\)circuit argument.
Finally, in the regimes \(\frac12\leq x^2\leq 1\) and \(a=b\leq c\leq a+b\), we use additional symmetries to extend existing renormalisation inequalities and establish logarithmic fluctuations. This is consistent with the conjecture that the scaling limit is the Gaussian free field.
[arXiv] 
Phase diagram of the AshkinTeller model, CMP, with Yacine Aoun and Moritz Dober
Communications in Mathematical Physics (CMP), Vol. 405 (2024), no. 37, 133
The AshkinTeller model is a pair of interacting Ising models and has two parameters: \(J\) is a coupling constant in the Ising models and \(U\) describes the strength of the interaction between them. In the ferromagnetic case \(J,U>0\) on the square lattice, we establish a complete phase diagram conjectured in physics in 1970s (by Kadanoff and Wegner, Wu and Lin, Baxter and others): when \(J<U\), the transitions for the Ising spins and their products occur at two distinct curves that are dual to each other; when \(J\geq U\), both transitions occur at the selfdual curve. All transitions are shown to be sharp using the OSSS inequality.
We use a finitecriterion argument and continuity to extend the result of Peled and the third author from a selfdual point to its neighborhood. Our proofs go through the randomcluster representation of the AshkinTeller model introduced by ChayesMachta and PfisterVelenik and we rely on couplings to FKpercolation.
[arXiv]CMP] 
Structure of Gibbs measures for planar FKpercolation and Potts models, PMP, with Ioan Manolescu
Probability and Mathematical Physics (PMP), Vol. 4 (2023), no. 2, 209–256
We prove that all Gibbs measures of the \(q\)state Potts model on \(\mathbb{Z}^2\) are linear combinations of the extremal measures obtained as thermodynamic limits under free or monochromatic boundary conditions. In particular all Gibbs measures are invariant under translations. This statement is new at points of firstorder phase transition, that is at \(T=T_{c}(q)\) when \(q>4\). In this case the structure of Gibbs measures is the most complex in the sense that there exist \(q+1\) distinct extremal measures.
Most of the work is devoted to the FKpercolation model on \(\mathbb{Z}^{2}\) with \(q\geq 1\), where we prove that every Gibbs measure is a linear combination of the free and wired ones. The arguments are nonquantitative and follow the spirit of the seminal works of Aizenman and Higuchi, which established the Gibbs structure for the twodimensional Ising model. Infiniterange dependencies in FKpercolation (i.e., a weaker spatial Markov property) pose serious additional difficulties compared to the case of the Ising model. For example, it is not automatic, albeit true, that thermodynamic limits are Gibbs. The result for the Potts model is then derived using the EdwardsSokal coupling and autoduality. The latter ingredient is necessary since applying the EdwardsSokal procedure to a Gibbs measure for the Potts model does not automatically produce a Gibbs measure for FKpercolation.
Finally, the proof is generic enough to adapt to the FKpercolation and Potts models on the triangular and hexagonal lattices and to the loop \(O(n)\) model in the range of parameters for which its spin representation is positively associated.
[arXivPMP] 
Macroscopic loops in the loop \(O(n)\) model via the XOR trick, Preprint, with Nicholas Crawford, Matan Harel, and Ron Peled
The loop \(O(n)\) model is a family of probability measures on collections of nonintersecting loops on the hexagonal lattice, parameterized by \((n,x)\), where \(n\) is a loop weight and \(x\) is an edge weight. Nienhuis predicts that, for \(0 \leq n \leq 2\), the model exhibits two regimes: one with short loops when \(x < x_c(n)\), and another with macroscopic loops when \(x \geq x_c(n)\), where \(x_c(n) = 1/\sqrt{2 + \sqrt{2n}}\).
In this paper, we prove three results regarding the existence of long loops in the loop \(O(n)\) model. Specifically, we show that, for some \(\delta >0\) and any \((n,x) \in [1,1+\delta) \times (1 \delta, 1]\), there are arbitrarily long loops surrounding typical faces in a finite domain. If \(n \in [1,1+\delta)\) and \(x \in (1\delta,1/\sqrt{n}]\), we can conclude the loops are macroscopic. Next, we prove the existence of loops whose diameter is comparable to that of a finite domain whenever \(n=1, x \in (1,\sqrt{3}]\); this regime is equivalent to part of the antiferromagnetic regime of the Ising model on the triangular lattice. Finally, we show the existence of noncontractible loops on a torus when \(n \in [1,2], x=1\).
The main ingredients of the proof are: (i) the `XOR trick': if \(\omega\) is a collection of short loops and \(\Gamma\) is a long loop, then the symmetric difference of \(\omega\) and \(\Gamma\) necessarily includes a long loop as well; (ii) a reduction of the problem of finding long loops to proving that a percolation process on an auxiliary graph, built using the ChayesMachta and EdwardsSokal geometric expansions, has no infinite connected components; and (iii) a recent result on the percolation threshold of BenjaminiSchramm limits of planar graphs.
[arXiv] 
On the transition between the disordered and antiferroelectric phases of the \(6\)vertex model , EJP, with Ron Peled
Electronic Journal of Probability (EJP), 2023
The symmetric sixvertex model with parameters \(a,b,c > 0\) is expected to exhibit different behavior in the regimes \(a+b < c\) (antiferroelectric), \(a−b<c\leq a+b\) (disordered) and \(a−b > c\) (ferroelectric). In this work, we study the way in which the transition between the regimes \(a+b=c\) and \(a+b < c\) manifests.
When \(a+b<c\), we show that the associated height function is localized and its extremal periodic Gibbs states can be parametrized by the integers in such a way that, in the \(n\)th state, the heights \(n\) and \(n+1\) percolate while the connected components of their complement have diameters with exponentially decaying tails. When \(a+b=c\), the height function is delocalized.
The proofs rely on the BaxterKellandWu coupling between the sixvertex and the randomcluster models and on recent results for the latter. An interpolation between free and wired boundary conditions is introduced by modifying cluster weights. Using triangular lattice contours (\(\mathbb{T}\)circuits), we describe another coupling for height functions that in particular leads to a novel proof of the delocalization at \(a+b=c\). Finally, we highlight a spin representation of the sixvertex model and obtain a coupling of it to the AshkinTeller model on \(\mathbb{Z}^2\) at its selfdual line \(\sinh 2J=e^{−2U}\). When \(J<\), we show that each of the two Ising configurations exhibits exponential decay of correlations while their product is ferromagnetically ordered.
[arXivEJP] 
Exponential decay in the loop \(O(n)\) model: \(n > 1, x<\tfrac{1}{\sqrt{3}}+\varepsilon(n)\), PRPR, with Ioan Manolescu
In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius, (2021), 455470
We show that the loop \(O(n)\) model on the hexagonal lattice exhibits exponential decay of loop sizes whenever \(n> 1\) and \(x<\tfrac{1}{\sqrt{3}}+\varepsilon(n)\), for some suitable choice of \(\varepsilon(n)>0\).
It is expected that, for \(n \leq 2\), the model exhibits a phase transition in terms of~\(x\), that separates regimes of polynomial and exponential decay of loop sizes. In this paradigm, our result implies that the phase transition for \(n \in (1,2]\) occurs at some critical parameter \(x_c(n)\) strictly greater than that \(x_c(1) = 1/\sqrt3\). The value of the latter is known since the loop \(O(1)\) model on the hexagonal lattice represents the contours of the spinclusters of the Ising model on the triangular lattice.
The proof is based on developing \(n\) as \(1+(n1)\) and exploiting the fact that, when \(x<\tfrac{1}{\sqrt{3}}\), the Ising model exhibits exponential decay on any (possibly non simplyconnected) domain. The latter follows from the positive association of the FKIsing representation.
[arXivPRPR] 
Uniform Lipschitz functions on the triangular lattice have logarithmic variations, CMP, with Ioan Manolescu
Communications in Mathematical Physics (CMP), vol. 381, 3, (2021), 11531221
Uniform integervalued Lipschitz functions on a domain of size \(N\) of the triangular lattice are shown to have variations of order \(\sqrt{\log N}\). The level lines of such functions form a loop \(O(2)\) model on the edges of the hexagonal lattice with edgeweight one. An infinitevolume Gibbs measure for the loop \(O(2)\) model is constructed as a thermodynamic limit and is shown to be unique. It contains only finite loops and has properties indicative of scaleinvariance: macroscopic loops appearing at every scale. The existence of the infinitevolume measure carries over to height functions pinned at the origin; the uniqueness of the Gibbs measure does not. The proof is based on a representation of the loop \(O(2)\) model via a pair of spin configurations that are shown to satisfy the FKG inequality. We prove RSWtype estimates for a certain connectivity notion in the aforementioned spin model.
[arXivCMP] 
Selfavoiding walk on \(\mathbb{Z}^2\) with YangBaxter weights: universality of critical fugacity and 2point function, AIHP, with Ioan Manolescu
Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques (AIHP), vol. 56, 4, (2020)
We consider a selfavoiding walk model (SAW) on the faces of the square lattice \(\mathbb{Z}^2\). This walk can traverse the same face twice, but crosses any edge at most once. The weight of a walk is a product of local weights: each square visited by the walk yields a weight that depends on the way the walk passes through it. The local weights are parametrised by angles \(\theta\in[\frac{\pi}{3},\frac{2\pi}{3}]\) and satisfy the YangBaxter equation. The selfavoiding walk is embedded in the plane by replacing the square faces of the grid with rhombi with corresponding angles.
By means of the YangBaxter transformation, we show that the 2point function of the walk in the halfplane does not depend on the rhombic tiling (i.e. on the angles chosen). In particular, this statistic coincides with that of the selfavoiding walk on the hexagonal lattice. Indeed, the latter can be obtained by choosing all angles \(\theta\) equal to \(\frac{\pi}{3}\).
For the hexagonal lattice, the critical fugacity of SAW was recently proved to be equal to \(1+\sqrt{2}\). We show that the same is true for any choice of angles. In doing so, we also give a new short proof to the fact that the partition function of selfavoiding bridges in a strip of the hexagonal lattice tends to 0 as the width of the strip tends to infinity. This proof also yields a quantitative bound on the convergence.
[arXivAIHP] 
Macroscopic loops in the loop \(O(n)\) model at Nienhuis' critical point, JEMS, with Hugo DuminilCopin, Ron Peled and Yinon Spinka
Journal of the European Mathematical Society (JEMS), vol. 23, 1, (2021), 315347
The loop \(O(n)\) model is a model for a random collection of nonintersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin \(O(n)\) model. It has been predicted by Nienhuis that for \(0\le n\le 2\) the loop \(O(n)\) model exhibits a phase transition at a critical parameter \(x_c(n)=\tfrac{1}{\sqrt{2+\sqrt{2n}}}\). For \(0<n\le 2\), the transition line has been further conjectured to separate a regime with short loops when \(x<x_c(n)\) from a regime with macroscopic loops when \(x\ge x_c(n)\).
In this paper, we prove that for \(n\in [1,2]\) and \(x=x_c(n)\) the loop \(O(n)\) model exhibits macroscopic loops. This is the first instance in which a loop \(O(n)\) model with \(n\neq 1\) is shown to exhibit such behaviour. A main tool in the proof is a new positive association (FKG) property shown to hold when \(n \ge 1\) and \(0<x\le\frac{1}{\sqrt{n}}\). This property implies, using techniques recently developed for the randomcluster model, the following dichotomy: either long loops are exponentially unlikely or the origin is surrounded by loops at any scale (boxcrossing property). We develop a 'domain gluing' technique which allows us to employ Smirnov's parafermionic observable to rule out the first alternative when \(x=x_c(n)\) and \(n\in[1,2]\).
[arXivJEMS] 
Discrete stressenergy tensor in the loop \(O(n)\) model, Preprint, with Dmitry Chelkak and Stanislav Smirnov
We study the loop \(O(n)\) model on the honeycomb lattice. By means of local nonplanar deformations of the lattice, we construct a discrete stressenergy tensor. For \(n\in [2,2]\), it gives a new observable satisfying a part of CauchyRiemann equations. We conjecture that it is approximately discreteholomorphic and converges to the stressenergy tensor in the continuum, which is known to be a holomorphic function with the Schwarzian conformal covariance. In support of this conjecture, we prove it for the case of \(n=1\) which corresponds to the Ising model. Moreover, in this case, we show that the correlations of the discrete stressenergy tensor with primary fields converge to their continuous counterparts, which satisfy the OPEs given by the CFT with central charge \(c=1/2\).
Proving the conjecture for other values of \(n\) remains a challenge. In particular, this would open a road to establishing the convergence of the interface to the corresponding \(\mathrm{SLE}_\kappa\) in the scaling limit.
[arXiv] 
On the probability that selfavoiding walk ends at a given point, AOP, with Hugo DuminilCopin, Alan Hammond and Ioan Manolescu
The Annals of Probability, 44(2) 955983 March 2016
We prove two results on the delocalization of the endpoint of a uniform selfavoiding walk on \(\mathbb{Z}^ {d}\) for \(d\geq2\). We show that the probability that a walk of length \(n\) ends at a point \(x\) tends to \(0\) as \(n\) tends to infinity, uniformly in \(x\). Also, when \(x\) is fixed, with \(\Vert x\Vert=1\), this probability decreases faster than \(n^{1/4+\varepsilon}\) for any \(\varepsilon>0\). This provides a bound on the probability that a selfavoiding walk is a polygon.
[arXivAOP] 
Electronic Communications in Probability (ECP) 20 (2015), no. 86, 113
We consider a selfavoiding walk on the dual \(\mathbb{Z}^2\) lattice. This walk can traverse the same square twice but cannot cross the same edge more than once. The weight of each square visited by the walk depends on the way the walk passes through it and the weight of the whole walk is calculated as a product of these weights. We consider a family of critical weights parametrized by angle \(\theta\in[\frac{\pi}{3},\frac{2\pi}{3}]\). For \(\theta=\frac{\pi}{3}\), this can be mapped to the selfavoiding walk on the honeycomb lattice. The connective constant in this case was proved to be equal to \(\sqrt{2+\sqrt{2}}\) by DuminilCopin and Smirnov. We generalize their result.
[arXivECP] 
(Russian) Zap. Nauchn. Sem. S.Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 417 (2013), Kombinatorika i Teoriya Grafov. VI, 1185;
Translation in J. Math. Sci. (N.Y.) 204 (2015), no. 2, 185–231
The work started in a previous paper is continued, and \(k\)cutsets in \(k\)connected graphs are studied. Several new statements concerning the structure of generalized flowers in \(k\)connected graphs are proved. Generalized flowers in the case \(k = 4\) are considered thereafter. For \(k = 4\) we give a description of the mutual arrangement of the intersections of two maximal generalized flowers with empty center, which have a common cutset.
[JMS] 
Forms of higher degree over certain fields, POMI, with Alexander Sivatski, Dmitry Stolyarov and Pavel Zatitsky
(Russian) Zap. Nauchn. Sem. S.Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 394 (2011), Voprosy Teorii Predstavlenii Algebr i Grupp. 22, 209217, 296;
Translation in J. Math. Sci. (N.Y.) 188 (2013), no. 5, 591–595
Let \(F\) be a nonformally real field, \(n, r\) be positive integers. Suppose that for any prime number \(p ≤ n\), the quotient group \(F^{*}/F^{*p}\) is finite. We prove that if \(N\) is large enough, then any system of \(r\) forms of degree in \(N\) variables over \(F\) has a nonzero solution. Also we show that if, in addition, \(F\) is infinite, then any diagonal form with nonzero coefficients of degree \(n\) in \(F^{*}/F^{*n}\) variables is universal, i.e,, its set of nonzero values coincides with \(F^{*}\)
[JMS] 
(Russian) Zap. Nauchn. Sem. S.Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 391 (2011), Kombinatorika i Teoriya Grafov. III, 4578;
Translation in J. Math. Sci. (N.Y.) 184 (2012), no. 5, 579–594
In this paper, we study \(k\)cutsets in \(k\)connected graphs. We introduce generalized flowers and prove some fundamental statements describing their structure. After that, we consider generalized flowers in the case where \(k = 4\). For \(k = 4\), we give a complete description of \(4\)cutsets belonging to a generalized flower.
[JMS]