Related papers: Lecture notes on "Combinatorial Criteria for Uniqu…
Much recent rigorous study of the classical ferromagnetic Ising model has been powered by its graphical representations, such as the random current and loop O(1) model (high temperature expansion). In this paper, we prove uniqueness of…
These are lecture notes of a course taken in Leipzig 2023, spring semester. It deals with extremal combinatorics, algebraic methods and combinatorial geometry. These are not meant to be exhaustive, and do not contain many proofs that were…
We relate the MNS and CKM mixing matrices using ideas from grand unification. We catalog models in terms of the family symmetries of the down quark mass matrices, and emphasize the role of the Cabibbo angle in the lepton mixing matrix. We…
These are the lecture notes of a course taught at the Park City Mathematics Institute in June 2017. They are intended to review some recent results, obtained in large part with Thomas Lebl\'e, on the statistical mechanics of systems of…
Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years,…
We introduce an statistical mechanical formalism for the study of discrete-time stochastic processes with which we prove: (i) General properties of extremal chains, including triviality on the tail $\sigma$-algebra, short-range…
This is a self-contained set of lecture notes covering various aspects of the theory of open quantum system, at a level appropriate for a one-semester graduate course. The main emphasis is on completely positive maps and master equations,…
This thesis is a multidisciplinary contribution to the information theory of single-particle Coulomb systems in their relativistic and not relativistic description, to the theory of special functions of mathematical physics with the…
In this paper, we describe several different meanings for the concept of Gibbs measure on the lattice $\mathbb{N}$ in the context of finite alphabets (or state space). We compare and analyze these "in principle" distinct notions: DLR-Gibbs…
We consider the phenomenological implications of a class of compactified string theories which naturally reproduces the flavour multiplet structure of the Standard Model. The implications for gauge unification depends on which of three…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
Strong typicality and the Markov lemma have been used in the proofs of several multiterminal source coding theorems. Since these two tools can be applied to finite alphabets only, the results proved by them are subject to the same…
The apparent unification of gauge couplings in Grand Unified Theories around 10$^{16}$ GeV is one of the strong arguments in favor of Supersymmetric extensions of the Standard Model. In this paper, an analysis of the measurements of the…
Gibbs random fields play an important role in statistics, for example the autologistic model is commonly used to model the spatial distribution of binary variables defined on a lattice. However they are complicated to work with due to an…
The aim of these notes is to provide a reasonably short and "hands-on" introduction to the differential calculus on associative algebras over a field of characteristic zero. Following a suggestion of Ginzburg's we call the resulting theory…
These notes were compiled as lecture notes for a course developed and taught at the University of the Southern California. They should be accessible to a typical engineering graduate student with a strong background in Applied Mathematics.…
Combinatorial enumeration leads to counting generating functions presenting a wide variety of analytic types. Properties of generating functions at singularities encode valuable information regarding asymptotic counting and limit…
Data reproducibility is a critical issue in all scientific experiments. In this manuscript, we consider the problem of quantifying the reproducibility of graphical measurements. We generalize the concept of image intra-class correlation…
The diverse quantization phenomena in 2D condensed-matter systems, being due to a uniform perpendicular magnetic field and the geometry-created lattice symmetries, are the focuses of this book. They cover the diversified magneto-electronic…
These notes aim to provide a pedagogical introduction to Lattice QCD. The topics covered include the scope of LQCD calculations, lattice discretization of gauge and fermion (naive, Wilson, and staggered) actions, doubling problem, improved…