ICTS

Planar Ising and bipartite dimer models are classical `free fermionic’ lattice systems: their partition functions can be written as Pfaffians of certain matrices that - in many setups - can be naturally interpreted as discrete Dirac operators. On $\mathbb Z^2$, this implies the discrete holomorphicity property of `fermionic observables’, that is, the matrix entries of the inverse operators, and leads to the classical proofs of conformal invariance of the critical Ising model (Smirnov) and of the dimer model in domains with special `Temperleyan’ boundaries (Kenyon). However, it is well known that the same set of discrete equations can give rise to very different complex structures that describe the limit of dimer fluctuations in domains with different boundaries, which makes the link between discrete holomorphicity and conformal invariance rather subtle. The situation is even less clear if the underlying graph and edge weights are irregular.

The main goal of this mini-course is to discuss a geometric procedure that aims to identify the `complex structure’ of a given graph equipped with the planar bipartite dimer (resp., Ising) model by associating with it a piecewise linear surface in the Minkowski space $\mathbb R^{2,2}$ (resp., $\mathbb R^{2,1}$), the so-called $t$- (resp., $s$-) surfaces. In particular, this framework unifies all known `linear’ notions of discrete holomorphicity. For the bipartite dimer model, it provides discrete complex analysis techniques in situations in which one expects that a conformal structure different from the standard Euclidean one appears in the limit. For the Ising model, this framework often allows one to work with various critical or near-critical setups using the same set of tools.

The behaviour of a random walk on a hyperbolic group is intimately connected to the geometry of the underlying space. In this talk, I will explain how random walk trajectories reflect the underlying geometry, and how they can serve as a canonical substitute for geodesics in discrete hyperbolic settings, where geodesics need not be unique.
    This is an example of why it is useful to study stochastic processes in relation to the underlying space. The geometry of the ambient space influences both the qualitative and quantitative features of the walk, and understanding this interaction helps clarify which aspects of the behaviour are specific to hyperbolic geometry and which may extend to other settings. This is based on joint work with Mahan Mj and Chiranjib Mukherjee.

A number of percolation models have recently witnessed considerable progress, notably in setups that are low-dimensional yet transient for the random walk (for instance, the 3-dimensional cubic lattice). The course will attempt to give an accessible introduction to this circle of ideas and present some recent results.

Planar Ising and bipartite dimer models are classical `free fermionic’ lattice systems: their partition functions can be written as Pfaffians of certain matrices that - in many setups - can be naturally interpreted as discrete Dirac operators. On $\mathbb Z^2$, this implies the discrete holomorphicity property of `fermionic observables’, that is, the matrix entries of the inverse operators, and leads to the classical proofs of conformal invariance of the critical Ising model (Smirnov) and of the dimer model in domains with special `Temperleyan’ boundaries (Kenyon). However, it is well known that the same set of discrete equations can give rise to very different complex structures that describe the limit of dimer fluctuations in domains with different boundaries, which makes the link between discrete holomorphicity and conformal invariance rather subtle. The situation is even less clear if the underlying graph and edge weights are irregular.

The main goal of this mini-course is to discuss a geometric procedure that aims to identify the `complex structure’ of a given graph equipped with the planar bipartite dimer (resp., Ising) model by associating with it a piecewise linear surface in the Minkowski space $\mathbb R^{2,2}$ (resp., $\mathbb R^{2,1}$), the so-called $t$- (resp., $s$-) surfaces. In particular, this framework unifies all known `linear’ notions of discrete holomorphicity. For the bipartite dimer model, it provides discrete complex analysis techniques in situations in which one expects that a conformal structure different from the standard Euclidean one appears in the limit. For the Ising model, this framework often allows one to work with various critical or near-critical setups using the same set of tools.

We study the spectra of matrix-valued contractions of the Gaussian Orthogonal Tensor Ensemble (GOTE). Let $\mathcal G$ denote a random tensor of dimension $n$ and order-$r$, drawn from the density
\[
    f(\mathcal G) = \frac{1}{Z_r(n)} \exp\bigg(-\frac{1}{2r}\|\mathcal G\|^2_{\mathrm{F}}\bigg).
\]
We consider contractions of the form $\mathcal G \cdot \mathbf w^{\otimes (r - 2)}$ when both $r$ and $n$ go to infinity such that $r / n \to c \in [0, \infty]$. We obtain a Baik-Ben Arous-P\'{e}ch\'{e} phase transition for the largest and the smallest eigenvalues of such contractions at $r = 3$. We also show that the extreme eigenvectors contain non-trivial information about $\mathbf w$. In fact, in the regime $1 \ll r \ll n$, there are two vectors, one of which is perfectly aligned with $w$. We also obtain some results on mixed contractions $\mathcal G \cdot \mathbf u \otimes \mathbf v$ in the case $r = 4$. This talk is based on a joint work with Soumendu Sundar Mukherjee.

Two-point functions of models like self-avoiding walk and percolation on $\mathbb Z^d$, in their subcritical regimes,exhibit so-called Ornstein--Zernike decay: at a subcritical parameter $z$ they decay at large distance like a multiple of $|x|_z^{-(d-1)/2} \exp[-m_z|x|_z]$ for some $m_z>0$ and some norm $|\cdot|_z$ on $\mathbb R^d$.  On the other hand, at the critical point $z=z_c$ and above the upper critical dimension (e.g., $4$ for self-avoiding walk), the decay is instead $\|x\|_2^{-(d-2)}$.
This course consists of two parts.

Part~I provides an introduction to the lace expansion for self-avoiding walk in dimensions $d>4$, and its application to prove critical decay of the form $\|x\|_2^{-(d-2)}$.  Convergence of the lace expansion is proved via the relatively very simple method of \emph{Ann. Inst.\ H.\ Poincar\'e Probab.\ Statist.}, 58, 26-33, 2023.
The method has been widely extended in recent papers with and by Yucheng Liu, also by Matthew Dickson and Yucheng Liu.
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Part~II concerns a new and general method for proving Ornstein--Zernike decay for random walk in all dimensions $d \ge 1$, self-avoiding walk for $d>4$, and related models, based on joint work in progress with Yucheng Liu. The new method makes explicit the cross-over from Ornstein--Zernike decay to critical decay as $z$ approaches the critical point $z_c$. These results greatly extend earlier work with Emmanuel Michta that proved a similar theorem for the specific example of simple random walk, via quite different methods, in \emph{ALEA, Lat.\ Am.\ J.\ Probab.\ Math.\ Stat.}, 19:957--981, 2022.
Previous literature mostly does not reveal the crossover.

Two-point functions of models like self-avoiding walk and percolation on $\mathbb Z^d$, in their subcritical regimes,exhibit so-called Ornstein--Zernike decay: at a subcritical parameter $z$ they decay at large distance like a multiple of $|x|_z^{-(d-1)/2} \exp[-m_z|x|_z]$ for some $m_z>0$ and some norm $|\cdot|_z$ on $\mathbb R^d$.  On the other hand, at the critical point $z=z_c$ and above the upper critical dimension (e.g., $4$ for self-avoiding walk), the decay is instead $\|x\|_2^{-(d-2)}$.
This course consists of two parts.

Part~I provides an introduction to the lace expansion for self-avoiding walk in dimensions $d>4$, and its application to prove critical decay of the form $\|x\|_2^{-(d-2)}$. Convergence of the lace expansion is proved via the relatively very simple method of \emph{Ann. Inst.\ H.\ Poincar\'e Probab.\ Statist.}, 58, 26-33, 2023. The method has been widely extended in recent papers with and by Yucheng Liu, also by Matthew Dickson and Yucheng Liu.
\\
Part~II concerns a new and general method for proving Ornstein--Zernike decay for random walk in all dimensions $d \ge 1$, self-avoiding walk for $d>4$, and related models, based on joint work in progress with Yucheng Liu. The new method makes explicit the cross-over from Ornstein--Zernike decay to critical decay as $z$ approaches the critical point $z_c$. These results greatly extend earlier work with Emmanuel Michta that  proved a similar theorem for the specific example of simple random walk, via quite different methods, in \emph{ALEA, Lat.\ Am.\ J.\ Probab.\ Math.\ Stat.}, 19:957--981, 2022.
Previous literature mostly does not reveal the crossover.

Tutte's barycentric (or harmonic) embeddings of planar graphs appear in various different contexts including combinatorics, electrostatics, discrete geometry and more. Recent studies of 2d lattice models suggest yet another angle from which Tutte's embeddings can be viewed as "discrete uniformizing maps": given an abstract planar graph (a planar map) with positive edge weights, we can hope that such embeddings encode the "discrete conformal structure" induced by the simple random walk on this graph. This poses the following question: given a sequence of Tutte barycentric embeddings, describe the diffusion that appears in the limit of the underlying simple random walks.

In this talk, I will discuss our recent joint work with Chelkak, Laslier and Russkikh where we study this problem using the theory of discrete complex analysis on t-embeddings. In this work we show that the aforementioned diffusion must be generated by the linearized Monge-Ampere equation whose potential is determined as a limit of Maxwell-Cremona lifts of the embeddings. We will discuss the convergence of Green's functions and solutions of Dirichlet problems, and how one can use t-embeddings to find a coordinate change making the diffusion to be a Brownian motion (if such a change exists).

In the last decades, FK percolation established itself as a central model within statistical mechanics, both for its relation to other models and for its intrinsic behavior. This minicourse will be dedicated to FK-percolation on the square lattice $\mathbb Z^2$ and will specifically touch on the sharpness of the phase transition, its continuity/discontinuity and properties of the critical phase. 
The ultimate goal is to prove that, for cluster-weights between 1 and 4 (for which the phase transition is continuous), the critical model exhibits invariance under rotations at large scales. This is indicative of the existence of a rotationally-invariant scaling limit, and potentially conformally invariant...

For a general graph, the compression exponent quantifies how efficiently the graph can be embedded into an $\ell^p$ space while controlling large-scale distortion. In this talk, we explore the connection between this notion and percolation theory. Concretely, our goal is to present a construction of percolations on general graphs that serves a dual purpose. First, we show how these percolations provide quantitative geometric information through their two-point functions and marginal parameters. Second, we explain how compression bounds can be recovered from the probabilistic data encoded in the constructed percolations. This correspondence establishes a new link between quantitative embedding theory and percolation on general graphs. (joint works with K. Recke)

In the last decades, FK percolation established itself as a central model within statistical mechanics, both for its relation to other models and for its intrinsic behavior. This minicourse will be dedicated to FK-percolation on the square lattice $\mathbb Z^2$ and will specifically touch on the sharpness of the phase transition, its continuity/discontinuity and properties of the critical phase. 
The ultimate goal is to prove that, for cluster-weights between 1 and 4 (for which the phase transition is continuous), the critical model exhibits invariance under rotations at large scales. This is indicative of the existence of a rotationally-invariant scaling limit, and potentially conformally invariant...

We report recent progress on the FK-Ising model on general (non-flat) $s$-embeddings with uniformly bounded geometry of faces. Namely, adapting the renormalization argument of Duminil-Copin, Manolescu, and Tassion to this setup, we prove the so-called `strong’ RSW-estimates that guarantee the existence of crossings in arbitrary topological rectangles with unfavorable boundary conditions.
In a special situation when the corresponding $s$-surfaces converge to a Lorenz-maximal surface in $\mathbb R^{2,1}$ this allows one to upgrade a recent result of S.C.Park on the convergence of interfaces to SLE curves to the convergence of loop ensembles to CLEs defined in the conformal structure of the limiting surface. The latter also implies the convergence of spin correlations normalized by one-point expectations due to known results of Camia and Feng.
Joint work with Yanqing Wei (UMichigan).

The class of one-component spin models with an even single-site potential is large and diverse. Depending on the shape of the potential, there is predicted to be a wide range of behaviour including no phase transition, discontinuous transitions or even multiple transitions. The most famous example is the Ising model, now known to undergo a second-order transition - expected to be indicative of a universality class of similar Ising behaviour. However, rigorous results in this direction have generally been restricted to the Ising model and its direct relatives, such as the $\phi^4$ model.
A central question is to extend these results and understand which other potentials will behave `like' the Ising model? 
 
We provide partial answers to this question by proving various results for a class of `double-well' potentials, including all polynomials with non-negative quartic and higher terms. In particular, the focus of this talk will be a new result on sharpness of the phase transition for models in this class. Other results include a characterisation of pffaffianity and lower bounds for the critical two-point function. The underlying tool is a new graphical representation for general spin models which can replace the use of the double random current and its switching lemma in many instances.

Joint work with Diederik van Engelenburg and Marcin Lis.

The talk will be about a natural percolation model built from the so-called Brownian loop soup. We will give sense to studying its phase transition in dimension $d = 2 + \varepsilon$, with $\varepsilon$ varying in $[0, 1]$, and discuss how to perform a rigorous ``epsilon-expansion'' in this context. Our approach gives access to a whole family of universality classes, and elucidate the behaviour of critical exponents etc. near the (lower-)critical dimension, which for this model is $d=2$. Based on joint works with Wen Zhang.

Entropic repulsion refers to when an ordinarily localized surface becomes delocalized in the presence of a floor. A classical setting to study this is the low temperature Ising model with Dobrushin boundary conditions: consider a box in $\mathbb Z^3$ with plus boundary conditions on the bottom side, and minus on the other five sides of the box. Then there is a surface separating the plus boundary from the minus boundary. Fr\"{o}hlich and Spencer proved in '87 that the height of this surface diverges; we prove it diverges logarithmically with the size of the box, and extend this to the Potts model. I will also discuss the open question of what the correct height of the surface should be up to leading order, resulting in somewhat paradoxical predictions. Based off joint works with Reza Gheissari and Eyal Lubetzky.

Percolation on sequences of finite graphs with growing size gives rise to many interesting phenomena that are different from those coming from percolation on infinite graphs. It is a very important problem to give criteria for specific bounded degree finite graphs to exhibit mean-field (near-) critical behavior (i.e., like the Erd\H{o}s-R\'enyi random graphs) and substantial research has been devoted to this. In this talk, we revisit this old problem by looking beyond the Bernoulli percolation and consider the level-set percolation of the zero-average Gaussian free field (GFF) on finite metric graphs. It turns out that for this model, one can deduce mean-field behavior for any family of graphs with spectral gap bounded away from 0 (which includes expanders, Hamming graphs, complete graphs etc.) --- a result unknown in such generality for Bernoulli percolation to the best of our knowledge. We will also discuss results on supercritical sharpness and possibilities to consider more general families of graphs. Based on joint works with Subhajit Goswami.

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 randomcurrent 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\'{u}lveda and Werner. Joint work with Tomás Alcalde L\'{o}pez and Lorca Heeney.

We show that the double random current representation of the Ising model satisfies local uniqueness of macroscopic cluster with high probability for every $\beta>\beta_c$, uniformly in boundary conditions (i.e. sources). Such a statement is the strongest version of what is sometimes called supercritical sharpness in percolation theory, and yields a detailed description of the model via standard renormalization techniques. In particular, this easily implies the exponential decay or truncated correlations for Ising and exponential mixing for the FK-Ising measure, results that had already been obtained by Duminil-Copin, Goswami and Raoufi. However, our proof is shorter and simpler, as we avoid the use of a highly technical multi-valued map principle by using a sprinkling argument instead.

Based on joint work with Trishen Gunaratnam, Christoforos Panagiotis and Romain Panis.

Consider 
(i) balls $\Lambda_n$ of growing volumes in the $d$-dimensional hierarchical lattice, and
(ii) the $d$-dimensional discrete torus $\mathbb{T}_n^d$ on $n^d$ vertices.
Place edges independently between each pair of vertices $x\neq y\in\Lambda_n$ or $\mathbb{T}_n^d$ with probability $1-\exp(-\beta J(x, y))$ where $J(x, y) \asymp \| x-y \|^{-\alpha}$ for some $0<\alpha<5d/6$.

We discuss recent results on the intrinsic geometry of these two models around the point of phase transition: Inside the critical window, the scaling limit of the maximal components viewed as metric measure spaces is Brownian, and these models belong to the Erdos-Renyi universality class when $0<\alpha<5d/6$. Further, when $0<\alpha<2d/3$, the girth of each maximal component in the critical window is $\Omega_P(|\Lambda_n|^{1/3})$ and $\Omega_P(n^{d/3})$ respectively for these two models, contrary to the situation when $d<\alpha\leq 4d/3$ where the girth would equal $3$.

It was recently conjectured by Hutchcroft that the model of critical hierarchical percolation with $\alpha\in(d, 4d/3]$ is a member of the Erdos-Renyi universality class, and we believe that this is also true for all $\alpha\in (0, d]$. These results, in particular, take a first step in that direction.

The study of critical phenomena in lattice statistical mechanical models such as percolation and spin systems has a long history in both physics and mathematics.  A central problem is to prove existence and calculate the values of the critical exponents that govern the universal behaviour of the model at and near its critical point.  Detailed proofs are typically available only for models in dimension two, or above an upper critical dimension where mean-field behaviour is observed.  We present a new, unified, generic, probabilistic, and relatively very simple proof of mean-field critical behaviour for high-dimensional models containing a small parameter.  The results apply to spin systems and self-avoiding walk in dimensions above 4, percolation in dimensions above 6, and lattice trees in dimensions above 8.  Minimal model-specific data is required for these applications.  This is joint work in progress with Hugo Duminil-Copin, Aman Markar and Romain Panis.

Once-reinforced random walk is a simple self-interacting process defined as follows. Edges that the walker already traversed at some point in the past are given a higher weight, 1+a for some a>0, compared to unvisited edges which are given weight 1. Sidoravicius' conjecture is that in dimension d>2 the process undergoes a phase transition in a. For small a it is diffusive and scales to Brownian motion, while for large a it is strictly subdiffusive and scales to Brownian motion reflected from the boundary of a slowly expanding balloon. We show the small a part of the conjecture, for d>5. Joint work with Dor Elboim.

In this talk, we will discuss the classical Ornstein-Zernike theory for the random-cluster model (also known as FK percolation). In its modern form, it is a very robust theory, which most celebrated output is the computation of the asymptotically polynomial corrections to the pure exponential decay of the two-points correlation function of the random-cluster model in the subcritical regime. We will present a recent work that extends this theory to the near-critical regime of the two-dimensional random-cluster model, thus providing a precise understanding of the Ornstein-Zernike asymptotics when $p$ approaches the critical parameter $p_c$. The output of this work is a formula encompassing both the critical behaviour of the system when looked at a scale negligible with respect to its correlation length, and its subcritical behaviour when looked at a scale way larger than its correlation length. Based on a joint work with Ioan Manolescu.

The double random current model is a graphical representation of the Ising model which has been a cornerstone in rigorous developments in the past forty years - in particular for the study of the Ising model in general dimension beyond the subcritical phase. However, unlike the more well-studied FK-percolation model, the random current lacks most of the monotonicity properties required to get a good handle on its intrinsic features. In particular, it was known for a while that the geometry of the infinite cluster in the random current model plays a big role in the continuity of the phase transition of the Ising model - more precisely, if the infinite cluster is one-ended, then the Ising phase transition is continuous. While the continuity question has been settled by other means, it nonetheless remains an interesting challenge in pure percolation to describe the geometry of the infinite cluster and in this talk, we will discuss a recent proof that the infinite cluster, when it exists, is, indeed, one-ended.

  Based on ongoing work with Frederik Ravn Klausen.

We investigate (via large-scale computational studies) percolation phenomena associated with the spatial extent of monomer-carrying regions of maximum-density dimer packings of site-diluted versions of non-bipartite lattices such as the triangular lattice and bipartite lattices such as the square lattice in two dimensions, as well as other examples in three dimensions. We find unusual percolated phases, some of which seem to violate thermodynamic self-averaging.