Alexander Glazman @ University of Innsbruck

Assoz. Prof. Alexander Glazman, PhD

Office 728, Technikerstraße 13/7, 6020 Innsbruck
Phone: +43 512 507 53930

About me

I am an Associate Professor at the University of Innsbruck and the PI of the FWF Stand-Alone grant "Order-disorder phase transitions in 2D lattice models".

My field of research lies at the intersection of Probability Theory and Mathematical Physics. Models of statistical mechanics that I am mostly interested in are:
Loop O(n) model, Six-vertex model, Ashkin-Teller model, random-cluster model, Self-Avoiding Walk, Ising model.

I did my Ph.D. at the University of Geneva in 2016 under the supervision of Stanislav Smirnov. The title of my PhD thesis is Properties of self-avoiding walks and a stress-energy tensor in the O(n) model. I obtained a degree Candidate of Physico-mathematical sciences and completed my master's degree in St Petersburg at PDMI and SPbU, both under supervision of Dmitry Karpov.

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

Discontinuous transition in 2D Potts: I. Order-disorder interface convergence, Preprint, with Moritz Dober and Sébastien Ott

It is known that the planar $q$-state Potts model undergoes a discontinuous phase transition when $q>4$ and there are exactly $q+1$ extremal Gibbs measures at the transition point: $q$ ordered (monochromatic) measures and one disordered (free). We focus on the Potts model under the Dobrushin order--disorder boundary conditions on a finite $N\times N$ part of the square grid. Our main result is that this interface is a well defined object, has $\sqrt{N}$ fluctuations, and converges to a Brownian bridge under diffusive scaling. The same holds also for the corresponding FK-percolation model for all $q>4$.
Our proofs rely on a coupling between FK-percolation, the six-vertex model, and the random-cluster representation of an Ashkin--Teller model (ATRC), and on a detailed study of the latter. The coupling relates the interface in FK-percolation to a long subcritical cluster in the ATRC model. For this cluster we develop a ``renewal picture'' à la Ornstein-Zernike. This is based on fine mixing properties of the ATRC model that we establish using the link to the six-vertex model and its height function. Along the way, we derive various properties of the Ashkin-Teller model, such as Ornstein-Zernike asymptotics for its two-point function.
In a companion work, we provide a detailed study of the Potts model under order-order Dobrushin conditions. We show emergence of a free layer of width $\sqrt{N}$ between the two ordered phases (wetting) and establish convergence of its boundaries to two Brownian bridges conditioned not to intersect.
[arXiv]
On loops in the complement to dimers, Preprint, with Lucas Rey

We consider ergodic translation-invariant Gibbs measures for the dimer model (i.e. perfect matchings) on the hexagonal lattice. The complement to a dimer configuration is a fully-packed loop configuration: each vertex has degree two. This is also known as the loop O(1) model at $x=\infty$. We show that, if the measure is non-frozen, then it exhibits either infinitely many loops around every face or a unique bi-infinite path.
Our main tool is the flip (or XOR) operation: if a hexagon contains exactly three dimers, one can replace them by the other three edges. Classical results in the dimer theory imply that such hexagons appear with a positive density. Up to some extent, this replaces the finite-energy property and allows to make use of tools from the percolation theory, in particular the Burton--Keane argument, to exclude existence of more than one bi-infinite path.
[arXiv]
Delocalisation and Continuity in 2D: Loop $O(2)$, Six-Vertex, and Random-Cluster Models, CMP, with Piet Lammers

accepted to Communications in Mathematical Physics
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 integer-valued Lipschitz function on the triangular lattice and settles one side of the conjecture of Fan, Domany, and Nienhuis: they predicted in the 1970s-80s that $x^2=\frac12$ is the critical point for the localisation-delocalisation transition.
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 $1\leq q\leq 4$. We rely neither on integrability tools (parafermionic observables, Bethe Ansatz), nor on the Russo-Seymour-Welsh theory.
Our approach goes through a novel FKG property that enables the use of the non-coexistence theorem of Zhang and Sheffield to prove delocalisation all the way up to the critical point. In the six-vertex 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 Ashkin-Teller model, CMP, with Yacine Aoun and Moritz Dober

Communications in Mathematical Physics (CMP), Vol. 405 (2024), no. 37, 1--33
The Ashkin-Teller 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 self-dual curve. All transitions are shown to be sharp using the OSSS inequality.
We use a finite-criterion argument and continuity to extend the result of Peled and the third author from a self-dual point to its neighborhood. Our proofs go through the random-cluster representation of the Ashkin-Teller model introduced by Chayes-Machta and Pfister-Velenik and we rely on couplings to FK-percolation.
[arXiv]|CMP]
Structure of Gibbs measures for planar FK-percolation 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 first-order 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 FK-percolation 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 non-quantitative and follow the spirit of the seminal works of Aizenman and Higuchi, which established the Gibbs structure for the two-dimensional Ising model. Infinite-range dependencies in FK-percolation (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 Edwards-Sokal coupling and auto-duality. The latter ingredient is necessary since applying the Edwards-Sokal procedure to a Gibbs measure for the Potts model does not automatically produce a Gibbs measure for FK-percolation.
Finally, the proof is generic enough to adapt to the FK-percolation 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.
[arXiv|PMP]
Macroscopic loops in the loop $O(n)$ model via the XOR trick, accepted to AOP, with Nicholas Crawford, Matan Harel, and Ron Peled

accepted to Annals of Probability (AOP)
The loop $O(n)$ model is a family of probability measures on collections of non-intersecting loops on the hexagonal lattice, parameterized by a loop-weight $n$ and an edge-weight $x$. Nienhuis predicts that, for $0 \leq n \leq 2$, the model exhibits two regimes separated by $x_c(n) = 1/\sqrt{2 + \sqrt{2-n}}$: when $x < x_c(n)$, the loop lengths have exponential tails, while, when $x \geq x_c(n)$, the loops are macroscopic.
In this paper, we prove three results regarding the existence of long loops in the loop $O(n)$ model:
-- In the regime $(n,x) \in [1,1+\delta) \times (1- \delta, 1]$ with $\delta >0$ small, a configuration sampled from a translation-invariant Gibbs measure will either contain an infinite path or have infinitely many loops surrounding every face. In the subregime $n \in [1,1+\delta)$ and $x \in (1-\delta,1/\sqrt{n}]$our results further imply Russo--Seymour--Welsh theory. This is the first proof of the existence of macroscopic loops in a positive area subset of the phase diagram.
-- 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.
-- Existence of non-contractible 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 planar graph, built using the Chayes--Machta and Edwards--Sokal geometric expansions, has no infinite connected components; and (iii) a recent result on the percolation threshold of Benjamini--Schramm 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 six-vertex 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 Baxter-Kelland-Wu coupling between the six-vertex and the random-cluster 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 six-vertex model and obtain a coupling of it to the Ashkin-Teller model on $\mathbb{Z}^2$ at its self-dual 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.
[arXiv|EJP]
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), 455-470
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 spin-clusters of the Ising model on the triangular lattice.
The proof is based on developing $n$ as $1+(n-1)$ and exploiting the fact that, when $x<\tfrac{1}{\sqrt{3}}$, the Ising model exhibits exponential decay on any (possibly non simply-connected) domain. The latter follows from the positive association of the FK-Ising representation.
[arXiv|PRPR]
Uniform Lipschitz functions on the triangular lattice have logarithmic variations, CMP, with Ioan Manolescu

Communications in Mathematical Physics (CMP), vol. 381, 3, (2021), 1153-1221
Uniform integer-valued 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 edge-weight one. An infinite-volume 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 scale-invariance: macroscopic loops appearing at every scale. The existence of the infinite-volume 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 RSW-type estimates for a certain connectivity notion in the aforementioned spin model.
[arXiv|CMP]
Self-avoiding walk on $\mathbb{Z}^2$ with Yang-Baxter weights: universality of critical fugacity and 2-point function, AIHP, with Ioan Manolescu

Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques (AIHP), vol. 56, 4, (2020)
We consider a self-avoiding 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 Yang-Baxter equation. The self-avoiding walk is embedded in the plane by replacing the square faces of the grid with rhombi with corresponding angles.
By means of the Yang-Baxter transformation, we show that the 2-point function of the walk in the half-plane does not depend on the rhombic tiling (i.e. on the angles chosen). In particular, this statistic coincides with that of the self-avoiding 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 self-avoiding 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.
[arXiv|AIHP]
Macroscopic loops in the loop $O(n)$ model at Nienhuis' critical point, JEMS, with Hugo Duminil-Copin, Ron Peled and Yinon Spinka

Journal of the European Mathematical Society (JEMS), vol. 23, 1, (2021), 315--347
The loop $O(n)$ model is a model for a random collection of non-intersecting 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 $x_c(n)=\tfrac{1}{\sqrt{2+\sqrt{2-n}}}$. 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 random-cluster model, the following dichotomy: either long loops are exponentially unlikely or the origin is surrounded by loops at any scale (box-crossing 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 $n\in[1,2]$.
[arXiv|JEMS]
Discrete stress-energy 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 non-planar deformations of the lattice, we construct a discrete stress-energy tensor. For $n\in [-2,2]$, it gives a new observable satisfying a part of Cauchy-Riemann equations. We conjecture that it is approximately discrete-holomorphic and converges to the stress-energy 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 stress-energy 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 self-avoiding walk ends at a given point, AOP, with Hugo Duminil-Copin, Alan Hammond and Ioan Manolescu

The Annals of Probability, 44(2) 955-983 March 2016
We prove two results on the delocalization of the endpoint of a uniform self-avoiding 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 self-avoiding walk is a polygon.
[arXiv|AOP]
Connective constant for a weighted self-avoiding walk on $\mathbb{Z}^2$, ECP

Electronic Communications in Probability (ECP) 20 (2015), no. 86, 1-13
We consider a self-avoiding 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 self-avoiding walk on the honeycomb lattice. The connective constant in this case was proved to be equal to $\sqrt{2+\sqrt{2}}$ by Duminil-Copin and Smirnov. We generalize their result.
[arXiv|ECP]
Generalized flowers in $k$-connected graph. Part 2, POMI

(Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 417 (2013), Kombinatorika i Teoriya Grafov. VI, 11-85;
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, 209--217, 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]
Generalized flowers in $k$-connected graph, POMI

(Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 391 (2011), Kombinatorika i Teoriya Grafov. III, 45-78;
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]