Related papers: Some Properties of the Plaquette Random-Cluster Mo…

For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric…

We introduce a 2-dimensional lattice model of granular matter. We use a combination of proof and simulation to demonstrate an order/disorder phase transition in the model, to which we associate the granular phenomenon of random close…

This paper studies cluster algebras locally, by identifying a special class of localizations which are themselves cluster algebras. A `locally acyclic cluster algebra' is a cluster algebra which admits a finite cover (in a geometric sense)…

For $\Pi \subset \mathbb{R}^2$ a connected, open, bounded set whose boundary is a finite union of disjoint polygons whose vertices have integer coordinates, the logarithm of the discrete Laplacian on $L\Pi \cap \mathbb{Z}^2$ with Dirichlet…

The problem of determining the volume of a tubular neighbourhood has a long and rich history. Bounds on the volume of neighbourhoods of algebraic sets have turned out to play an important role in the probabilistic analysis of condition…

Given two polynomials $p(x), q(x)$ of degree $d$, we give a combinatorial formula for the finite free cumulants of $p(x)\boxtimes_d q(x)$. We show that this formula admits a topological expansion in terms of non-crossing multi-annular…

We describe a quantitative construction of almost-normal diffeomorphisms between embedded orientable manifolds with boundary to be used in the study of geometric variational problems with stratified singular sets. We then apply this…

We study the critical behavior of the q-state Potts model with random ferromagnetic couplings. Working with the cluster representation the partition sum of the model in the large-q limit is dominated by a single graph, the fractal…

A closed subscheme of codimension two $T \subset P^2$ is a quasi complete intersection (q.c.i.) of type $(a,b,c)$ if there exists a surjective morphism $\mathcal{O} (-a) \oplus \mathcal{O} (-b) \oplus \mathcal{O} (-c) \to \mathcal{I} _T$.…

In the Constrained-degree percolation model on a graph $(\mathbb{V},\mathbb{E})$ there are a sequence, $(U_e)_{e\in\mathbb{E}}$, of i.i.d. random variables with distribution $U[0,1]$ and a positive integer $k$. Each bond $e$ tries to open…

The technique of $Q$-polinomials are used to derive the $w$- constraints in the two-matrix and Kontsevich-like model at finite $N$. These constraints are closed and form Lie algebra. They are associated with the matrices, $\lambda…

We define a stochastic lattice model for a fluctuating directed polymer in $d\geq 2$ dimensions. This model can be alternatively interpreted as a fluctuating random path in 2 dimensions, or a one-dimensional asymmetric simple exclusion…

In this paper, we focus on a new lower bound quantum cluster algebra which is generated by the initial quantum cluster variables and the quantum projective cluster variables of an acyclic quantum cluster algebra with principal coefficients.…

We propose a new class of models for variable clustering called Asymptotic Independent block (AI-block) models, which defines population-level clusters based on the independence of the maxima of a multivariate stationary mixing random…

Let $Q$ be any invertible valued quiver without oriented cycles. We study connections between the category of valued representations of $Q$ and expansions of cluster variables in terms of the initial cluster in quantum cluster algebras. We…

We study a random partial covering model on the $(d-1)$-dimensional unit sphere, where $N$ spherical caps are placed independently and uniformly at random, each covering a surface fraction of $1/N$. This model provides a continuous…

We explain the relation between certain random tiling models and interacting particle systems belonging to the anisotropic KPZ (Kardar-Parisi-Zhang) universality class in 2+1-dimensions. The link between these two \emph{a priori} disjoint…

Cluster algebras are a recent topic of study and have been shown to be a useful tool to characterize structures in several knowledge fields. An important problem is to establish whether or not a given cluster algebra is of finite type.…

We provide a complete classification of the singularities of cluster algebras of finite type with trivial coefficients. Alongside, we develop a constructive desingularization of these singularities via blowups in regular centers over fields…

In this paper we consider a probability distribution on plane partitions, which arises as a one-parameter generalization of the q^{volume} measure. This generalization is closely related to the classical multivariate Hall-Littlewood…