This is Oberwolfach’s mini workshop 2047b.

Various models of two-dimensional statistical mechanics are expected to feature conformally invariant objects in the scaling limit at criticality, such as Schramm-Loewner evolutions and the Gaussian free field. While such proofs exists for some models on particular lattices, a major challenge is to achieve universality, namely to build a framework which encompasses arbitrary lattices with arbitrary weights. Important tools in this area include discrete complex analysis and the geometry of algebraic curves (the spectral curves associated with some models).

This workshop will focus on three models: dimers, Ising and spanning trees. Starting twenty years ago, these models have been studied for special classes of weights, called critical isoradial weights. These arise from embeddings of the lattice as circle patterns where all the circles have unit radius and correspond to spectral curves of genus zero.

Recent years have witnessed progress in two directions. The first one is the resolution of inverse spectral problem in positive genus by Fock, George, Goncharov and Kenyon describing the correspondence between spectral curves with marked points and models on periodic graphs.

Some ingredients of this correspondence can be used to construct, from a spectral curve given a priori, Gibbs measures for these models on non periodic graphs, which would benefit from locality properties. This locality in turns allows for explicit computation of thermodynamical quantities for these measures. This has been established for genus 1 curves by Boutillier, de Tilière, Cimasoni and Raschel, and is work in progress for higher genus, generalizing the results of Kenyon.

The second is the introduction of universal graph embeddings that generalize isoradial circle patterns: s-embeddings for the Ising model by Chelkak and Lis and t-embeddings for the dimer model by Affolter, Chelkak, Kenyon, Lam, Laslier, Ramassamy and Russkikh.

These universal embeddings translate the local moves of the statistical mechanical models into incidence theorems in planar geometry, which are governed by cluster algebra mutations.

The next major challenges include applying the aforementioned results to derive universal scaling limits results and connecting the two-dimensional graph embeddings to non-Euclidean three-dimensional objects. A more distant yet crucial goal would be to use these universal embeddings to study the scaling limits of random planar maps.

In November 2019, a series of talks during a workshop at Banff introduced t-embeddings and their potential applications to study dimer scaling limits and raised a lot of interest and open questions. One year after would be the perfect time to assess progress made so far.

- Cédric Boutillier (Paris, France)
- Sanjay Ramassamy (Gif-sur-Yvette, France)
- Marianna Russkikh (Cambridge, USA)

- Niklas Affolter (Berlin, Germany)
- Nathanaël Berestycki (Wien, Austria)
- Cédric Boutillier (Paris, France)
- Dmitry Chelkak (Paris, France)
- David Cimasoni (Geneva, Switzerland)
- Béatrice de Tilière (Paris, France)
- Vladimir Fock (Strasbourg, France)
- Terrence George (Ann Arbor, USA)
- Nahid Ghodratipour (Tehran, Iran)
- Richard Kenyon (New Haven, USA)
- Benoı̂t Laslier (Paris, France)
- Zhongyang Li (Storrs, USA)
- Marcin Lis (Vienna, Austria)
- Paul Melotti (Fribourg, Switzerland)
- Asaf Nachmias (Tel Aviv, Israel)
- Sanjay Ramassamy (Gif-sur-Yvette, France)
- Marianna Russkikh (Cambridge, USA)

**Be careful**: indicated times are for the (winter) **Central Europe time zone**
(meaning UTC+01). Please check the corresponding local time. Every talk lasts for 50 minutes, followed by 10 minutes of questions.

- Monday 16 November
- 1:50pm – Opening remarks and introduction of all the participants
- 2:00pm – Nathanaël Berestycki, Dimers with free boundary: random walk representation and scaling limit
- 3:00pm – Marcin Lis, On boundary correlations in the Ashkin—Teller model
- 4:00pm – Break
- 4:30pm – Zhongyang Li, Percolation in the hyperbolic plane and Benjamini-Schramm conjecture

- Tuesday 17 November
- 2:00pm – Dmitry Chelkak, S-/t-/p-embeddings: state-of-the-art of convergence results for the bipartite dimer and Ising models on irregular planar graphs part 1 (geogebra file)
- 3:00pm – Break
- 3:30pm – Vladimir Fock, Tau-function on Riemann surfaces
- 4:30pm – discussions

- Wednesday 18 November
- 2:00pm – David Cimasoni, Elliptic dimer models and genus 1 Harnack curves
- 3:00pm – Richard Kenyon, The Multidimer Model
- 4:00pm – Break
- 4:30pm – Niklas Affolter, Triple crossing diagrams, projective configurations, dimers

- Thursday 19 November
- 2:00pm – Dmitry Chelkak, S-/t-/p-embeddings: state-of-the-art of convergence results for the bipartite dimer and Ising models on irregular planar graphs part 2
- 3:00pm – Break
- 3:30pm – Asaf Nachmias, The Dirichlet problem for orthodiagonal maps
- 4:30pm – discussions

- Friday 20 November
- 2:00pm – Paul Melotti, Cube flips in s-embeddings and alpha-embeddings
- 3:00pm – Dmitry Chelkak, S-/t-/p-embeddings: state-of-the-art of convergence results for the bipartite dimer and Ising models on irregular planar graphs part 3
- 4:00pm – Break
- 4:30pm – Terrence George, Inverse spectral problem for biperiodic planar networks

Title: S-/t-/p-embeddings: state-of-the-art of convergence results for the bipartite dimer and Ising models on irregular planar graphs.

Plan of the lectures:

In these three lectures, based upon recent joint works with Benoît Laslier, Sanjay Ramassamy and Marianna Russkikh, as well as upon ongoing discussions with colleagues, I plan to describe a new viewpoint on convergence results for

- height fluctuations in the bipartite dimer model embedded into the complex plane using a Coulomb gauge (the notion introduced by Kenyon, Lam, Ramassamy and Russkikh) aka a t-embedding;
- particular setup of perfect t-embeddings=p-embeddings;
- specification of this construction to the Ising model aka s-embeddings.

The goal of this series of talks is to provide a brief overview of the following topics, not going into details of the proofs:

- Basics on discrete holomorphic functions on t-embeddings: definitions, a priori regularity theory.
- Relevance of 2+1 and 2+2 Minkowski spaces, and of minimal surfaces in these spaces; (massive) holomorphicity of fermionic observables in the `small mesh size’ limit.
- Convergence to the GFF in the setup of p-embeddings converging to a minimal surface; state-of-the-art of related results for the Ising model.
- Open questions (e.g., the existence of p-embeddings) and perspectives.

I hope to benefit from the traditional spirit of mini-workshops at the MFO and to keep the presentation open to questions and remarks.

**Nathanaël Berestycki**,*Dimers with free boundary: random walk representation and scaling limit*

We study the dimer model in which particles on part of the boundary are allowed to form monomers with some fixed weight $z>0$ called the monomer fugacity. A bijection relates this model to a non-bipartite dimer, whose corresponding Kasteleyn matrix describes a random walk with ‘‘negative rates’’ along the boundary. Yet under certain assumptions on the domain boundary, we prove an effective random walk representation for the inverse Kasteleyn matrix. This allows us to show first that the infinite volume (thermodynamic) limit exists on the upper half plane. Second, we show in that case that the scaling limit of the height function is given by a multiple of the Gaussian free field with free (or Neumann) boundary conditions, thereby answering a question of Giuliani et al. To our knowledge, this is the first occurrence of this field as a scaling limit in the study of height functions.

Joint work with Marcin Lis (Vienna) and Wei Qian (Paris).

**Marcin Lis**,*On boundary correlations in the Ashkin—Teller model*

In this talk I will describe recent results that generalize the switching lemma of Griffiths, Hurst and Sherman to the random current representation of the Ashkin-Teller model. I will use this 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. I will 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.

**Zhongyang Li**,*Percolation in the hyperbolic plane and Benjamini-Schramm conjecture*

I will show that for a non-amenable, locally finite, connected, transitive, planar graph with one end, any automorphism invariant site percolation on the graph does not have exactly 1 infinite 1-cluster and exactly 1 infinite 0-cluster a.s. If we further assume that the site percolation is insertion-tolerant and a.s. there exists a unique infinite 0-cluster, then a.s. there are no infinite 1-clusters. I will also discuss how to apply these results to solve two conjectures of Benjamini and Schramm in 1996.

**Vladimir Fock**,*Tau-function on Riemann surfaces*

Tau-functions of Sato are certain generating functions for solution of integrable PDE like KdV, Sine-Gordon etc. There are many versions of these functions and usually the definition is quite complicated and uses semi-infinite forms. We are going to give a more direct definition for a tau-function as an algebro-geometric object. We will show also that a tau function gives a solution to a cluster integrable system and give an explicit formula in terms of theta functions.

**David Cimasoni**,*Elliptic dimer models and genus 1 Harnack curves*

The aim of this talk is to present a joint project with Cédric Boutillier and Béatrice de Tilière about the dimer model on minimal graphs with Fock’s elliptic weights. I will start by introducing the principal characters, namely minimal graphs and elliptic weights, before explaining our main results. The first one is an explicit local expression for a two-parameter family of inverses of the corresponding Kasteleyn operator. When the minimal graph satisfies a natural condition, we then construct a family of dimer Gibbs measures from these inverses and describe the associated phase diagram. Finally, in the periodic case, we establish a correspondence between these models and Harnack curves of genus 1.

**Richard Kenyon**,*The Multidimer Model*

This is joint work with Andrei Pohoata. Given a graph G, let G_N be obtained by replacing each vertex with N vertices and each edge with K_{N,N}. We consider the dimer model on G_N. We compute the free energy in the limit of large N. We relate the fluctuations of the edge process to the GFF on G. We solve an analog of the monomer/dimer problem in this setting, showing an exact Coulomb gas limit.

**Niklas Affolter**,*Triple crossing diagrams, projective configurations, dimers*

We introduce TCD maps, which associate projective configurations to triple crossing diagrams. We obtain two dimer models (with complex edge and face weights) associated to every TCD map. The first dimer model is based on projective geometry and the second one on affine geometry. The 2-2 moves in the TCDs correspond to spider moves in the associated dimer models. We can characterize projective configurations that correspond to the resistor resp. Ising subvariety. In this way, we reproduce as special cases many of the known examples: s-, t- and harmonic embeddings, triangulations in CP^1, T-graphs, the pentagram map, dSKP lattices. We also obtain new dimer models for objects of discrete differential geometry, including Q-nets, Darboux maps and Line complexes. We show that the resistor subvariety appears naturally in relation to a symmetric bilinear form, while the Ising subvariety appears naturally in relation to an anti-symmetric bilinear form.

Joint work with Max Glick, Pavlo Pylyavskyy, Sanjay Ramassamy

**Asaf Nachmias**,*The Dirichlet problem for orthodiagonal maps*

We prove that the discrete harmonic function corresponding to smooth Dirichlet boundary conditions on orthodiagonal maps, that is, plane graphs having quadrilateral faces with orthogonal diagonals, converges to its continuous counterpart as the mesh size goes to 0. This provides a convergence statement for discrete holomorphic functions, similar to the one obtained by Chelkak and Smirnov for isoradial graphs. By the double circle packing theorem, any finite, simple, 3-connected planar map admits an orthodiagonal representation and our result has no other “regularity” assumptions (such as bounded vertex degrees). Hence it is applicable to random planar maps models. Joint work with O. Gurel-Gurevich and D. Jerison.

**Paul Melotti**,*Cube flips in s-embeddings and alpha-embeddings*

In various models, a key to integrability is the existence of a local transformation or coupling of the underlying graph and parameters, such that long range properties remain unchanged. This may take the form of a “cube flip” (or star-triangle move) for Ising or spanning tree models, an urban renewal for dimers, etc. On the other hand, an intense line of research concerns the quest for canonical embeddings of such models (such as s-embeddings introduced by Chelkak for Ising models, t-embeddings for dimers, etc.). Through this correspondence, cube flips of the abstract model should only affect the embedding locally, and thus be conjugated to theorems of planar (or projective) geometry. I will present a few cases where this correspondence can be proved, and other families of embeddings with such a geometric property. Joint work with Sanjay Ramassamy and Paul Thévenin.

**Terrence George**,*Inverse spectral problem for biperiodic planar networks*

A biperiodic planar network is a pair (G, c) where G is a graph embedded on the torus and c is a function called conductance from the edges of G to non-zero complex numbers. Associated to the discrete Laplacian on a biperiodic planar network is its spectral transform: a spectral curve and a divisor on it. We show that the divisor is a point in the Prym variety of the curve and describe the inverse map along the lines of Fock’s inverse map for dimers. When the spectral curve has genus 0, we recover the isoradial conductances.