Related papers: A lecture on cluster expansions
This Meeting featured the recent advancements in our understanding of galaxy clusters and the distant Universe, achieved by the past and new generation of X-ray satellites. I summarize here the main themes that have been discussed: (a)…
We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of `extended quivers' which are oriented hypergraphs. We describe mutations of such…
This is a set of introductory lectures on the behaviour of a directed polymer in a random medium. Both the intuitive picture that helps in developing an understanding and systematic approaches for quantitative studies are discussed.
We introduce and analyse a class of fragmentation-coalescence processes defined on finite systems of particles organised into clusters. Coalescent events merge multiple clusters simultaneously to form a single larger cluster, while…
The depletion theory of nanoparticles immersed in a semidilute polymer solution is reinterpreted in terms of depleted chains of polymer segments. Limitations and extensions of mean-field theory are discussed. An explicit expression for the…
In gaussian theories of structure formation, the galaxy cluster abundance is an extremely sensitive probe of the density fluctuation power spectrum and of the density parameter, $\Omega$. We develop this theme by deriving and studying in…
The objective of the present paper is to prove cluster multiplication theorem in the quantum cluster algebra of type $A_{2}^{(2)}$. As corollaries, we obtain bar-invariant $\mathbb{Z}[q^{\pm\frac{1}{2}}]$-bases established in [6], and…
These lecture notes, adapted from the habilitation thesis of the author, survey in a first part various exact results obtained in the past few decades about KPZ fluctuations in one dimension, with a special focus on finite volume effects…
We study quantum cluster algebras from unpunctured surfaces with arbitrary coefficients and quantization. We first give a new proof of the Laurent expansion formulas for commutative cluster algebras from unpunctured surfaces, we then give…
This PhD deals with the notion of pseudo algebraically closed (PAC) extensions of fields. It develops a group-theoretic machinery, based on a generalization of embedding problems, to study these extensions. Perhaps the main result is that…
A central problem in many-body quantum physics is the determination of the ground state of a thermodynamically large physical system. We construct a cluster expansion for ground states of local Hamiltonians, which naturally incorporates…
These lecture notes provide an introduction to quantum cluster methods for strongly correlated systems. Cluster Perturbation Theory (CPT), the Variational Cluster Approximation (VCA) and Cellular Dynamical Mean Field Theory (CDMFT) are…
We are learning much about how structure forms, in particular how clusters as nodes in the cosmic web evolve and accrete matter, and about the physical processes within these objects. In the next decade, the study of clusters will enable us…
This draft is intended to be used as class notes for a grad course on rigorous statistical mechanics at math department of UFMG. It should be considered as a very prelimivary version and a work in progress. Several chapters lack references,…
In a cluster algebra, a subset of initial cluster variables can be specialised in such a way that all elements of the resulting algebra become polynomial in the remaining variables.
We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings.…
The structural and thermodynamic properties of mixtures of colloidal spheres and non-adsorbing polymer chains are studied within a novel general two-component macromolecular liquid state approach applicable for all size asymmetry ratios.…
The Atomic Cluster Expansion (ACE) [R. Drautz, Phys. Rev. B, 99:014104 (2019)] provides a systematically improvable, universal descriptor for the environment of an atom that is invariant to permutation, translation and rotation. ACE is…
The purpose of this lecture is to describe the KAM theorem in its most basic form and to give a complete and detailed proof. This proof essentially follows the traditional lines laid out by the inventors of this theory, and the emphasis is…
The aim of this paper is to study co-prolongations of central extensions. We construct the obstruction theory for co-prolongations and classify the equivalence classes of these by kernels of a homomorphisms between 2-dimensional cohomology…