Title: Some rigorous results on the Lévy spin glass model

Abstract: The Lévy spin glass model, proposed by Cizeau-Bouchaud, is a mean-field model defined on a fully connected graph, where the spin interactions are formulated through a power-law distribution. This model is well-motivated from the study of the experimental metallic spin glasses. It is also expected to bridge between some mean-field and diluted models. In this talk, we will discuss some recent progress on the Lévy model including its high temperature behavior and the existence and variational expression for the limiting free energy. Based on a joint work with Heejune Kim and Arnab Sen.

Abstract: Yang-Mills (YM) models arise in physics in the description of the fundamental forces and can be seen as non-Abelian extensions of Maxwell's theory of electromagnetism. In this talk, I will present YM models from the perspective of statistical mechanics, focusing on their definition on the lattice. The scaling limit of these lattice models was shown in 2 dimensions by T. Lévy and is known as the YM probability measure. I will present a recent result that establishes the invariance of the YM measure on the two-dimensional torus for the associated renormalised Langevin dynamic. Based on joint work with Hao Shen.

Tuesday, Nov 29, 2022: Konstantinos Kavvadias (Cambridge) 

Conformal removability of SLE_4 

We consider the Schramm-Loewner evolution (SLE_kappa) with kappa = 4, the critical value of kappa>0 at or below which SLE_kappa is a simple curve and above which it is self-intersecting. We show that the range of an SLE_4 curve is a.s. conformally removable, answering a question posed by Sheffield. In order to establish this result, we give a new sufficient condition for a set X in the complex plane to be conformally removable which applies in the case that X is not necessarily the boundary of a simply connected domain. This is based on a recent joint work with Jason Miller and Lukas Schoug 

Tuesday, Nov 29, 2022: Konstantinos Kavvadias (Cambridge) 

Conformal removability of SLE_4 

We consider the Schramm-Loewner evolution (SLE_kappa) with kappa = 4, the critical value of kappa>0 at or below which SLE_kappa is a simple curve and above which it is self-intersecting. We show that the range of an SLE_4 curve is a.s. conformally removable, answering a question posed by Sheffield. In order to establish this result, we give a new sufficient condition for a set X in the complex plane to be conformally removable which applies in the case that X is not necessarily the boundary of a simply connected domain. This is based on a recent joint work with Jason Miller and Lukas Schoug 

In this talk I will present one transformation that appears in very different contexts: combinatorial maps that get simplified, constellations that lose colours, cumulants that get freed, and x and y that get symplectically exchanged in topological recursion. I will explain how to realise all these dualities through a transformation that involves monotone Hurwitz numbers and we call master relation. Expressing the transformation as the action of an operator on the Fock space allows us to find functional relations that relate the generating series of higher order free cumulants and moments, which solves an open problem in free probability and generalises the R transform machinery of Voiculescu. I will also present the universal procedure of topological recursion, which appears in many algebro-geometric contexts; in particular, I will explain how it governs combinatorial maps and fully simple maps. Finally, this leads us to introduce a notion of surfaced free cumulants that captures the all-order asymptotic expansion in 1/N of random ensembles of matrices of size N in presence of some unitary invariance.

This is based on joint work with Gaëtan Borot, Séverin Charbonnier, Felix Leid and Sergey Shadrin.

I will survey different models of random hyperbolic surfaces of large genus, the analogies with random graphs that guide their study, and some of what is known. This will include my recent joint work with Mike Lipnowski (arXiv:2103.07496) that uses the Selberg trace formula and results of Mirzakhani to show the first eigenvalue of the Laplacian is typically large. 

I will explain how to implement conformal field theory in a simply-connected domain as a boundary version of theory on the Riemann sphere using the Schottky double construction. The fields in this theory are the statistical fields generated by background charge modification of the Gaussian free field with Dirichlet boundary conditions under the OPE multiplications. I will show that the correlation functions of such fields with symmetric background charges form a collection of martingale-observables for (forward) chordal/radial SLE with force points and spins. I will also present the connection between conformal field theory with Neumann boundary condition and the theory of backward SLE. Based on joint work with N. Makarov. 

Abstract: We prove that large Boltzmann stable planar maps of index $\alpha \in (1;2)$ converge  in the scaling limit towards a 

random compact metric space $ \mathcal{S}_{\alpha}$ that we construct explicitly. They form a one-parameter family of random continuous spaces ``with holes'' or ``faces''  different from the Brownian sphere.  In the so-called dilute phase $ \alpha \in [3/2;2)$, the topology of $ \mathcal{S}_{\alpha}$ is that of the Sierpinski carpet, while in the dense phase $ \alpha \in (1;3/2)$ the ``faces'' of $ \mathcal{S}_{\alpha}$ may touch each-others. En route, we prove various geometric properties of these objects concerning their faces or their geodesics.

Joint work with Grégory Miermont and Armand Riera.

Abstract:  This talk will consist basically of one naive question: could there be an interesting probabilistic story combining the 2D GFF, imaginary chaos, two-valued sets and c = 1 conformal field theories? 

I will try to explain the known connections and discuss some questions which might or might not be interesting, and might or might not be feasible.

Joint work with Christophe Garban. The discrete Coulomb gas is a model where an integer amount of charged particles are put on the $d$-dimensional grid. In this talk, I will discuss some basic properties of the Coulomb gas, its connection with other statistical physics models and the scaling limit of the potential of the discrete Coulomb gas at high enough temperature. 

Abstract: We consider critical planar site percolation on the triangular lattice and derive sharp estimates on asymptotics of the probability of half-plane $j$-arm events for $j\geq 1$ and whole-plane (polychromatic) $j$-arm events for $j>1$ under some specific boundary conditions. We also obtain up-to-constant estimates for other boundary conditions in the whole-plane case. These estimates greatly improve previous ones and solve a problem of Schramm (Proc. of ICM, 2006). In the course of proof, we also obtain a super-strong separation lemma, which confirms a conjecture by Garban, Pete and Schramm (J. Amer. Math. Soc., 2013) and is of independent interest. This is joint work in progress with Hang Du (PKU), Yifan Gao (PKU) and Zijie Zhuang (UPenn).

Abstract: In a groundbreaking work, Duplantier, Miller and Sheffield showed that subcritical Liouville quantum gravity (LQG) coupled with Schramm-Loewner evolutions (SLE) can be described by the mating of two continuum random trees. In this talk I will discuss the counterpart of their result for critical LQG and SLE. More precisely, I will explain how, as we approach criticality from the subcritical regime, the space-filling SLE degenerates to the uniform CLE_4 exploration introduced by Werner and Wu, together with a collection of independent coin tosses indexed by the branch points of the exploration. Furthermore, although the pair of continuum random trees collapse to a single continuum random tree in the limit we can apply an appropriate affine transform to the encoding Brownian motions before taking the limit, and get convergence to a Brownian half-plane excursion. I will try to explain how observables of interest in the critical CLE decorated LQG picture are encoded by a growth fragmentation naturally embedded in the Brownian excursion. This talk is based on joint work with Juhan Aru, Nina Holden and Xin Sun. 

If lengths 1 and 2 are assigned randomly to each edge in Z^2, what are

the fluctuations of distances between far away points?

This problem is open, yet we know, in great detail, what to expect. The

directed landscape, a universal random plane geometry, provides the

answer to such questions.

The double random current (DRC) model is a natural percolation model whose geometric properties are intimately related to spin correlations of the Ising model. In two dimensions, it moreover carries an integer valued height function on the graph, called the nesting field.

We study the critical DRC model on bounded domains of the square lattice. We fully describe the joint scaling limit of the (primal and dual) DRC clusters and the nesting field as the lattice mesh size vanishes. We prove that the nesting field becomes the Dirichlet Gaussian free field (GFF) in this limit, and that the outer boundaries of the DRC clusters with free boundary conditions are the conformal loop ensemble with $\kappa=4$ (CLE4) coupled to that GFF. Moreover, we also show that the inner boundaries of the DRC clusters form a two-valued local set with values ${\mp 2\lambda, (2\sqrt2 \mp 2) \lambda}$ for the field restricted to a CLE4 loop with boundary value $\pm 2\lambda$.

Our proof is a combination of exact solvability of the Ising model, new crossing estimates for the DRC model (which does not posses the FKG property), and a careful analysis of the structure of two-valued local sets of the continuum GFF.

This is joint work with Hugo Duminil-Copin and Wei Qian.

Conformal invariance of critical lattice models in two-dimensional has been vigorously studied for decades. The first example where the conformal invariance was rigorously verified was the planar uniform spanning tree (together with loop-erased random walk), proved by Lawler, Schramm and Werner in 2004. Later, the conformal invariance was also verified for Bernoulli percolation (Smirnov 2001), level lines of Gaussian free field (Schramm-Sheffield 2009), and Ising model and FK-Ising model (Chelkak-Smirnov et al 2012). In this talk, we focus on crossing probabilities of these critical lattice models in polygons with alternating boundary conditions. 

The talk has two parts. In the first part, we consider critical Ising model and give crossing probabilities of multiple interfaces in the critical Ising model in polygon with alternating boundary conditions. Similar formulas also hold for other models, for instance level lines of Gaussian free field and Bernoulli percolation. However, the situation is different when one considers uniform spanning tree. In the second part, we discuss uniform spanning tree and explain the corresponding results.

On the one hand, the 2D Gaussian free field (GFF) is a log-correlated Gaussian field whose exponential defines a random measure: the multiplicative chaos associated to the GFF, often called Liouville measure. On the other hand, the Brownian loop soup is an infinite collection of loops distributed according to a Poisson point process of intensity \theta times a loop measure. At criticality (\theta = 1/2), its occupation field is distributed like half of the GFF squared (Le Jan's isomorphism). 

The purpose of this talk is to understand the infinitesimal contribution of one loop to Liouville measure in the above coupling. This work is not restricted to the critical intensity and provides the natural notion of multiplicative chaos associated to the Brownian loop soup when \theta is not equal to 1/2.

Based on a joint work with É. Aïdékon, N. Berestycki and T. Lupu. https://arxiv.org/abs/2107.13340

First passage percolation (FPP) on Z^d or R^d is a canonical model of a random metric space where the standard Euclidean geometry is distorted by random noise. Of central interest is the length and the geometry of the geodesic, the shortest path between points. Since the latter, owing to its length minimization, traverses through atypically low values of the underlying noise variables, it is an important problem to quantify the disparity between the environment rooted at a point on the geodesic and the typical one. 

We will discuss recent work, done jointly with R. Basu and M. Bhatia, investigating this in the context of gamma-Liouville Quantum Gravity (LQG) (where gamma in (0,2) is a parameter) which can be thought of as a canonical model of a random two dimensional Riemannian surface. In particular, we consider the unique infinite geodesic Gamma from the origin, and show that, for an almost sure realization of the underlying noise given by a Gaussian Free Field, the distributions of the appropriately scaled field and the induced metric on a ball, rooted at a point “uniformly” sampled on Gamma (under a certain log-parametrization), converge to deterministic measures on the space of generalized functions and continuous metrics on the unit disk respectively. Towards a better understanding of the limiting objects, we will present results comparing them to their  typical counterparts.  

Time permitting, we will also review recent advances around this question for other models of random planar metrics known to exhibit different universal features.

Abstract: The Schramm-Loewner evolution (SLE) is a one-parameter family

of random planar fractal curves, which has been of considerable interest

since their introduction by Schramm in 1999, as they arise as scaling

limits in several two-dimensional statistical mechanics models at

criticality. Choosing the parameter $\kappa$ to be either 4 or 8 results

in special behaviour, as $\kappa = 4$ ($\kappa = 8) is the largest

(resp. smallest) $\kappa$ such that SLE$_\kappa$ curves are simple

(resp. space-filling). As such, regularity results in those cases differ

significantly from the cases of other values of $\kappa$. We will

discuss recent results on the modulus of continuity of the SLE$_4$

uniformizing map and the SLE$_8$ trace, as well as a byproduct of our

analysis, concerning the conformal removability of SLE$_4$. The talk is

based on joint work with Konstantinos Kavvadias and Jason Miller. 

Abstract: Consider an infinite planar graph with uniform polynomial growth of degree d > 2. Many examples of such graphs exhibit similar geometric and spectral properties, and it has been conjectured that this is necessary. We present a family of counterexamples. In particular, we show that for every rational d > 2, there is a planar graph with uniform polynomial growth of degree d on which the random walk is transient, disproving a conjecture of Benjamini (2011). By a well-known theorem of Benjamini and Schramm, such a graph cannot be a unimodular random graph. We also give examples of unimodular random planar graphs of uniform polynomial growth with unexpected properties. For instance, graphs of (almost sure) uniform polynomial growth of every rational degree d > 2 for which the speed exponent of the walk is larger than 1/d, and in which the complements of all balls are connected. This resolves negatively two questions of Benjamini and Papasoglou (2011).

Based on a joint work with James R. Lee.

Abstract: Liouville quantum gravity (LQG) was introduced by Polyakov as a canonical model of random surfaces. We explore the interplay between two facets of LQG: Liouville conformal field theory (LCFT) as constructed by the path integral approach, and the coupling of LQG with Schramm-Loewner evolution called mating-of-trees theory. We compute the mating-of-trees variance via LCFT. We also compute the one-point bulk structure constant of boundary LCFT via mating-of-trees, i.e. rigorously verify the explicit expression of Fateev, Zamolodchikov and Zamolodchikov. The key ingredients of our proofs are: 1. a unifying perspective which we call the uniform embedding of quantum surfaces; 2. conformal welding of these surfaces with SLE as their interfaces.

Based on a joint work with Nina Holden and Xin Sun, and a joint work with Guillaume Remy and Xin Sun.

Abstract: We introduce the stochastic Ricci flow in two spatial dimensions. The flow is symmetric with respect to a measure induced by Liouville Conformal Field Theory. Using the theory of Dirichlet forms, we construct a weak solution to the associated equation of the area measure on compact surfaces, in the L1 regime. 

I would also like to mention a few open questions and possible directions, and try to discuss with the audience about the connection between this “dynamical” approach and the other approaches of Liouville quantum gravity. Based on joint work with Julien Dubedat.

Abstract: In this talk, I will focus on the connection between Liouville

conformal field theory (LCFT) and the classical Liouville field theory via

the semiclassical approach. LCFT depends on a parameter \gamma \in (0,2)

and the limit \gamma \to 0 corresponds to the semiclassical limit of the

theory. Within this asymptotic and under a negative curvature condition

(on the limiting metric of the theory), one can determine the limit of the

correlation functions and of the associated Liouville field. I will also

present a large deviation result for the Liouville field: as expected, the

large deviation functional is the classical Liouville action.  As a

corollary, this yields a new (probabilistic) proof of the Takhtajan-Zograf

theorem which relates the classical Liouville action (taken at its

minimum) to Poincaré's accessory parameters. Finally, I will present

conjectures in the positive curvature case (including the study of the

so-called quantum spheres introduced by Duplantier-Miller-Sheffield).

Based on a joint work with H. Lacoin and R. Rhodes.

Laughlin state is a wave function for a configurations of N

interacting electrons in the plane, which is famous for accurately describing the fractional

quantum Hall effect (FQHE). As a mathematical object, it is customary to consider

the Laughlin states on a Riemann surface, where physical quantities can be understood as

a « response» to an arbitrary genus, moduli, conformal metric etc.

Besides giving a correct value for the Hall conductance in FQHE, it was understood some time ago

that they also have a bulk central charge and exhibit gravitational anomaly.

This poses the question whether (and how) the Laughlin states can be understood in terms of

quantum field theory and even 2d quantum gravity, when the number of particles tends to infinity.

I will attempt to overview recent physics and math progress on this aspect of the L. s. 

In this talk, I will discuss    Graeme Segal’s axioms for a Conformal Field Theory (CFT) and explain why they are valid in Liouville CFT. This allows us to express the n-point correlation function of vertex operators on a Riemann surface of arbitrary genus in terms of probabilistically defined amplitudes corresponding to punctured discs, annuli and pairs of pants. As a consequence we obtain a formula for the correlation functions as an integral associated to a pant  decomposition of the surface. The integrand is the modulus squared of a conformal block depending holomorphically in the moduli of the surface with marked points. The integration measure involves a product of DOZZ factors. This is ongoing work with C. Guillarmou, A. Kupiainen and V. Vargas. 

For a number of critical lattice models in 2D statistical physics, it has been proven that scaling limits of interfaces (with suitable boundary conditions) are described by Schramm-Loewner evolution (SLE) curves. So-called partition functions of these SLEs (which also encode macroscopic crossing probabilities) can be regarded as specific correlation functions in the conformal field theory (CFT) associated to the lattice model in question. Although it is not clear how to define the latter mathematically, one can still make sense of many of the properties predicted for these CFTs. In particular, all of the expected CFT properties: conformal invariance, null-field equations, and fusion rules, are satisfied by the partition functions. One might then ask: Is it possible to go deeper and to construct the appropriate CFT fields as random distributions? Time permitting, I discuss some ideas to this direction. 

Color each vertex of an infinite graph blue with probability p and red with probability 1-p, independently among vertices. For which values of p is there an infinite connected component of blue vertices? The talk will focus on this classical percolation problem for the class of planar graphs. Recently, Itai Benjamini made several conjectures in this context, relating the percolation problem to the behavior of simple random walk on the graph. We will explain how partial answers to Benjamini's conjectures may be obtained using the theory of circle packings. Among the results is the fact that the critical percolation probability admits a universal lower bound for the class of recurrent plane triangulations.

In long-range percolation on Z^d, each potential edge {x,y} is included independently at random with probability roughly β||x-y||^{-d-α}, where α > 0 controls how long-range the model is and β > 0 is an intensity parameter. The smaller α is, the easier it is for very long edges to appear. We are normally interested in fixing α and studying the phase transition that occurs as β is increased and an infinite cluster emerges. Perhaps surprisingly, the phase transition for long-range percolation is much better understood than that of nearest neighbour percolation, at least when α is small: It is a theorem of Noam Berger that if α < d then the phase transition is continuous, meaning that there are no infinite clusters at the critical value of β. (Proving the analogous result for nearest neighbour percolation is of course a notorious open problem!) In my talk I will describe a new, quantitative proof of Berger's theorem that yields power-law upper bounds on the distribution of the cluster of the origin at criticality. As a part of this proof, I will describe a new universal inequality stating that on any graph, the maximum size of a percolation cluster is of the same order as its median with high probability. 

I will talk about our recent results on all geodesics in the Brownian map, including those between exceptional points. 

First, we prove a strong and quantitative form of the confluence of geodesics phenomenon which states that any pair of geodesics which are sufficiently close in the Hausdorff distance must coincide with each other except near their endpoints.

Then, we show that the intersection of any two geodesics minus their endpoints is connected, the number of geodesics which emanate from a single point and are disjoint except at their starting point is at most 5, and the maximal number of geodesics which connect any pair of points is 9. For each k=1,…,9, we obtain the Hausdorff dimension of the pairs of points connected by exactly k geodesics. For k=7,8,9, such pairs have dimension zero and are countably infinite. Further, we classify the (finite number of) possible configurations of geodesics between any pair of points, up to homeomorphism, and give a dimension upper bound for the set of endpoints in each case. 

Finally, we show that every geodesic can be approximated arbitrarily well and in a strong sense by a geodesic connecting typical points.  In particular, this gives an affirmative answer to a conjecture of Angel, Kolesnik, and Miermont that the geodesic frame, the union of all of the geodesics in the Brownian map minus their endpoints, has dimension one, the dimension of a single geodesic.  

This is based on joint work with Jason Miller (https://arxiv.org/abs/2008.02242).

Abstract: One would like to construct useful probability measures on a large class of Riemannian metrics, e.g. on a conformal class of metrics on a fixed surface M.  The  approach of my talk is to approximate the infinite dimensional space of metrics by finite dimensional spaces B_k of metrics known as Bergman metrics of degree k.These  metrics arise from embeddings of M into projective spaces by holomorphic maps. B_k can be identified with positive Hermitian matrices  P_k of rank N_k \simeq k. We then define sequences  \mu_k of probability measures on B_k.  There are many ways to do this, some natural from a geometric viewpoint (Mabuchi metric, Calabi metric) and some natural from the identification with P_k. I will describe several \mu_k and what we know about their limit behavior.

Abstract: Liouville theory is a fundamental example of a conformal field theory (CFT) first introduced by A. Polyakov in the context of string theory. In recent years it has been rigorously studied using probabilistic techniques. In this talk we will study the integrable structure of Liouville CFT on a domain with boundary by proving exact formulas for its correlation functions. We will also explain connections with SLE curves. Based on joint work with Morris Ang, Xin Sun and Tunan Zhu.

Abstract: A theorem of Spitzer states that the Cauchy process on the circle can be obtained as the trace of Brownian motion in the disc. I will present an analogue of this result in the context of Liouville quantum gravity (LQG), the random geometry associated with the exponential of the Gaussian free field. There is a canonical Markov process on LQG which was constructed by Garban, Rhodes & Vargas (and also Berestycki): the so-called Liouville Brownian motion. I will introduce the boundary version of this process, the Cauchy process associated with LQG length. Using conformal welding (Sheffield's quantum zipper), I will then discuss some applications to Schramm-Loewner evolution.

Abstract: I will talk about the recent progress in understanding the nearest-neighbor Ising (resp. bipartite dimer) model on large irregular planar graphs via the so-called s- (resp., t-) embeddings of these graphs into the Minkowski R^{2+1} (resp., R^{2+2}) spaces. If such embeddings converge - in the `small mesh size' limit - to a Lorentz-minimal surface S, then the scaling limit of the model demonstrates the conformally invariant behavior; the conformal structure is inherited to S from the ambient space. However, if S is not minimal, then massive fermions appear; the mass is equal to the mean curvature of S at a given point. Though we are still quite far from justifying/understanding the whole picture and many things remain under construction, certain pieces were already put on a solid ground recently; see arXiv:2006:14559, arXiv:2001.11871, arXiv:2002.07540 + work in progress with Benoit Laslier and Marianna Russkikh.

I plan to give a slightly informal talk and to focus on presenting the key concepts as well as the state-of-the-art of relevant theorems and conjectures rather than on discussing details of the proofs. Hopefully, one day these developments could also lead to a new perspective for understanding the limits of random maps carrying the `critical' Ising (resp., bipartite dimer) model but this part remains very vague and purely speculative at the moment.

Abstract: I will be interested in the simple random walk on infinite random

planar maps. More precisely, I will present a new proof of the recurrence

of the simple random walk on the Uniform Infinite Half-Plane Map. The

argument is based on a self-duality property of the model and shares

similarities with Russo--Seymour--Welsh theory in percolation. It is not as

robust as other methods, but we obtain logarithmic lower bounds on the

resistances, which is better than what was previously known on comparable

models. This is based on joint work with Thomas Lehéricy

(https://arxiv.org/abs/1912.08790).

Abstract: In this lecture series, I will overview the large deviation results of SLEs in both the kappa to 0 and kappa to infinity regimes. The rate functions (referred to as various energies) turn out to be interesting deterministic functionals on more regular (quasiconformal) and more classical curves/surfaces/evolution families in geometric function theory. Moreover, the energy minimizers are smoother than generic finite energy objects as being analytic or even algebraic. 

I will show a close analogy between many couplings in random conformal geometry and identities on the rate functions in the more regular finite-energy world where proofs are often short and straightforward. The analogy may keep inspiring new directions to investigate on both sides.

Abstract: We study the Gaussian free field on the Euclidean lattice in three and more dimensions and consider the percolation problem associated with its level-sets, as first investigated by Lebowitz and Saleur in 1986. We prove the equality of several natural critical parameters associated with this model. Our findings yield the sharpness of the associated phase transition, which corresponds to classical results in the context of Bernoulli percolation due to Menshikov and Aizenman-Barsky (in the subcritical phase) and Grimmett-Marstrand (in the supercritical phase).  To the best of our knowledge, the only instances in which an analogous result is known to hold, in all dimensions greater or equal to three, are the random cluster representation of the Ising model and the Bernoulli percolation.  Based on joint work with H. Duminil-Copin, Pierre-F Rodriguez and Franco Severo.

Abstract: A long line of work in anomalous diffusion on fractals and self-similar random media confirms that the "Einstein relations" hold whenever the random walk is strongly recurrent (i.e., spectral dimension d_s < 2).  These are relations between scaling exponents that connect the mean displacement and return probabilities of the random walk to the density and conductivity of the underlying medium.

I will show that, on unimodular random networks, these relations hold somewhat more generally.  Most importantly, this includes the case d_s=2, capturing a rich class of random networks that are not strongly recurrent.  These include many models of random planar networks whose geometry can be derived from an exponentiated 2D Gaussian free field (e.g., the uniform infinite planar triangulation).  On the other hand, there is an example (from joint work with Ebrahimnejad) showing that the relations can fail for d_2 > 2 even in a unimodular random planar graph.

References:

Relations between scaling exponents in unimodular random graphs (https://arxiv.org/abs/2007.06548)

On planar graphs of uniform polynomial growth (https://arxiv.org/abs/2005.03139)