Combinatorics

Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization, Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions. This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session

Combinatorics

Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combinatorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session

On discrete surfaces : Enumerative geometry, matrix models and universality classes via topological recursion

The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. We approach our study of these objects from different perspectives, namely bijective combinatorics, matrix models and analysis of critical behaviors. Our problems have a powerful relatively recent tool in common, which is the so-called topological recursion introduced by Chekhov, Eynard and Orantin around 2007. Further understanding general properties of this procedure also constitutes a motivation for us. We introduce the notion of fully simple maps, which are maps with non self-intersecting disjoint boundaries. In contrast, maps where such a restriction is not imposed are called ordinary. We study in detail the combinatorial relation between fully simple and ordinary maps with topology of a disk or a cylinder. We show that the generating series of simple disks is given by the functional inversion of the generating series of ordinary disks. We also obtain an elegant formula for cylinders. These relations reproduce the relation between (first and second order) correlation moments and free cumulants established by Collins--Mingo--'Sniady--Speicher in the setting of free probability, and implement the exchange transformation $x leftrightarrow y$ on the spectral curve in the context of topological recursion. These interesting features motivated us to investigate fully simple maps, which turned out to be interesting combinatorial objects by themselves. We then propose a combinatorial interpretation of the still not well understood exchange symplectic transformation of the topological recursion. We provide a matrix model interpretation for fully simple maps, via the formal hermitian matrix model with external field. We also deduce a universal relation between generating series of fully simple maps and of ordinary maps, which involves double monotone Hurwitz numbers. In particular, (ordinary) maps without internal faces -- which are generated by the Gaussian Unitary Ensemble -- and with boundary perimeters $(lambda_1,ldots,lambda_n)$ are strictly monotone double Hurwitz numbers with ramifications $lambda$ above $infty$ and $(2,ldots,2)$ above $0$. Combining with a recent result of Dubrovin--Liu--Yang--Zhang, this implies an ELSV-like formula for these Hurwitz numbers. Later, we consider ordinary maps endowed with a so-called $O(mathsf{n})$ loop model, which is a classical model in statistical physics. We consider a probability measure on these objects, thus providing a notion of randomness, and our goal is to determine which shapes are more likely to occur regarding the nesting properties of the loops decorating the maps. In this context, we call volume the number of vertices of the map and we want to study the limiting objects when the volume becomes arbitrarily large, which can be done by studying the generating series at dominant singularities. An important motivation comes from the conjecture that the geometry of large random maps is universal. We pursue the analysis of nesting statistics in the $O(mathsf{n})$ loop model on random maps of arbitrary topologies in the presence of large and small boundaries, which was initiated for maps with the topology of disks and cylinders by Borot--Bouttier--Duplantier. For this purpose we rely on topological recursion results for the enumeration of maps in the $O(mathsf{n})$ model. We characterize the generating series of maps of genus $g$ with $k$ boundaries and~$k'$ marked points which realize a fixed nesting graph, which is associated to every map endowed with loops and encodes the information regarding non-separating loops, which are the non-contractible ones on the complement of the marked elements. These generating series are amenable to explicit computations in the so-called loop model with bending energy on triangulations, and we characterize their behavior at criticality in the dense and in the dilute phases, which are the two universality classes characteristic of the $O(mathsf{n})$ loop model. We extract interesting qualitative conclusions, e.g., which nesting graphs are more probable to occur. We also argue how this analysis can be generalized to other problems in enumerative geometry satisfying the topological recursion, and apply our method to study the fully simple maps introduced in the first part of the thesis

Structural Design and Analysis of Low-Density Parity-Check Codes and Systematic Repeat-Accumulate Codes

The discovery of two fundamental error-correcting code families, known as turbo codes and low-density parity-check (LDPC) codes, has led to a revolution in coding theory and to a paradigm shift from traditional algebraic codes towards modern graph-based codes that can be decoded by iterative message passing algorithms. From then on, it has become a focal point of research to develop powerful LDPC and turbo-like codes. Besides the classical domain of randomly constructed codes, an alternative and competitive line of research is concerned with highly structured LDPC and turbo-like codes based on combinatorial designs. Such codes are typically characterized by high code rates already at small to moderate code lengths and good code properties such as the avoidance of harmful 4-cycles in the code's factor graph. Furthermore, their structure can usually be exploited for an efficient implementation, in particular, they can be encoded with low complexity as opposed to random-like codes. Hence, these codes are suitable for high-speed applications such as magnetic recording or optical communication. This thesis greatly contributes to the field of structured LDPC codes and systematic repeat-accumulate (sRA) codes as a subclass of turbo-like codes by presenting new combinatorial construction techniques and algebraic methods for an improved code design. More specifically, novel and infinite families of high-rate structured LDPC codes and sRA codes are presented based on balanced incomplete block designs (BIBDs), which form a subclass of combinatorial designs. Besides of showing excellent error-correcting capabilites under iterative decoding, these codes can be implemented efficiently, since their inner structure enables low-complexity encoding and accelerated decoding algorithms. A further infinite series of structured LDPC codes is presented based on the notion of transversal designs, which form another subclass of combinatorial designs. By a proper configuration of these codes, they reveal an excellent decoding performance under iterative decoding, in particular, with very low error-floors. The approach for lowering these error-floors is threefold. First, a thorough analysis of the decoding failures is carried out, resulting in an extensive classification of so-called stopping sets and absorbing sets. These combinatorial entities are known to be the main cause of decoding failures in the error-floor region over the binary erasure channel (BEC) and additive white Gaussian noise (AWGN) channel, respectively. Second, the specific code structures are exploited in order to calculate conditions for the avoidance of the most harmful stopping and absorbing sets. Third, powerful design strategies are derived for the identification of those code instances with the best error-floor performances. The resulting codes can additionally be encoded with low complexity and thus are ideally suited for practical high-speed applications. Further investigations are carried out on the infinite family of structured LDPC codes based on finite geometries. It is known that these codes perform very well under iterative decoding and that their encoding can be achieved with low complexity. By combining the latest findings in the fields of finite geometries and combinatorial designs, we generate new theoretical insights about the decoding failures of such codes under iterative decoding. These examinations finally help to identify the geometric codes with the most beneficial error-correcting capabilities over the BEC

Probabilistic and extremal studies in additive combinatorics

The results in this thesis concern extremal and probabilistic topics in number theoretic settings. We prove sufficient conditions on when certain types of integer solutions to linear systems of equations in binomial random sets are distributed normally, results on the typical approximate structure of pairs of integer subsets with a given sumset cardinality, as well as upper bounds on how large a family of integer sets defining pairwise distinct sumsets can be. In order to prove the typical structural result on pairs of integer sets, we also establish a new multipartite version of the method of hypergraph containers, generalizing earlier work by Morris, Saxton and Samotij.L'objectiu de la combinatòria additiva “històricament també anomenada teoria combinatòria de nombres” és la d’estudiar l'estructura additiva de conjunts en determinats grups ambient. La combinatòria extremal estudia quant de gran pot ser una col·lecció d'objectes finits abans d'exhibir determinats requisits estructurals. La combinatòria probabilística analitza estructures combinatòries aleatòries, identificant en particular l'estructura dels objectes combinatoris típics. Entre els estudis més celebrats hi ha el treball de grafs aleatoris iniciat per Erdös i Rényi. Un exemple especialment rellevant de com aquestes tres àrees s'entrellacen és el desenvolupament per Erdös del mètode probabilístic en teoria de nombres i en combinatòria, que mostra l'existència de moltes estructures extremes en configuracions additives utilitzant tècniques probabilistes. Tots els temes d'aquesta tesi es troben en la intersecció d'aquestes tres àrees, i apareixen en els problemes següents. Solucions enteres de sistemes d'equacions lineals. Els darrers anys s'han obtingut resultats pel que fa a l’existència de llindars per a determinades solucions enteres a un sistema arbitrari d'equacions lineals donat, responent a la pregunta de quan s'espera que el subconjunt aleatori binomial d'un conjunt inicial de nombres enters contingui solucions gairebé sempre. La següent pregunta lògica és la següent. Suposem que estem en la zona en que hi haurà solucions enteres en el conjunt aleatori binomial, com es distribueixen aleshores aquestes solucions? Al capítol 1, avançarem per respondre aquesta pregunta proporcionant condicions suficients per a quan una gran varietat de solucions segueixen una distribució normal. També parlarem de com, en determinats casos, aquestes condicions suficients també són necessàries. Conjunts amb suma acotada. Què es pot dir de l'estructura de dos conjunts finits en un grup abelià si la seva suma de Minkowski no és molt més gran que la dels conjunts? Un resultat clàssic de Kneser diu que això pot passar si i només si la suma de Minkowski és periòdica respecte a un subgrup propi. En el capítol 3 establirem dos tipus de resultats. En primer lloc, establirem les anomenades versions robustes dels teoremes clàssics de Kneser i Freiman. Robust aquí es refereix al fet que en comptes de demanar informació estructural sobre els conjunts constituents amb el coneixement que la seva suma és petita, només necessitem que això sigui vàlid per a un subconjunt gran passa si només volem donar una informació estructural per a gairebé tots els parells de conjunts amb una suma d'una mida determinada? Donem un teorema d'estructura aproximat que s'aplica a gairebé la majoria dels rangs possibles per la mida dels conjunts suma. Sistemes de conjunts de Sidon. Les preguntes clàssiques sobre els conjunts de Sidon són determinar la seva mida màxima o saber quan un conjunt aleatori és un conjunt de Sidon. Al capítol 4 generalitzem la noció de conjunts de Sidon per establir sistemes i establim els límits corresponents per a aquestes dues preguntes. També demostrem un resultat de densitat relativa, resultat condicionat a una conjectura sobre l'estructura específica dels sistemes màxims de Sidon. Conjunts independents en hipergrafs. El mètode dels contenidors d'hipergrafs és una eina general que es pot utilitzar per obtenir resultats sobre el nombre i l'estructura de conjunts independents en els hipergrafs. La connexió amb la combinatòria additiva apareix perquè molts problemes additius es poden codificar com l'estudi de conjunts independents en hipergrafs.Postprint (published version

Three lectures on random proper colorings of Zd\mathbb{Z}^d

A proper $q$-coloring of a graph is an assignment of one of $q$ colors to each vertex of the graph so that adjacent vertices are colored differently. Sample uniformly among all proper $q$-colorings of a large discrete cube in the integer lattice $\mathbb{Z}^d$. Does the random coloring obtained exhibit any large-scale structure? Does it have fast decay of correlations? We discuss these questions and the way their answers depend on the dimension $d$ and the number of colors $q$. The questions are motivated by statistical physics (anti-ferromagnetic materials, square ice), combinatorics (proper colorings, independent sets) and the study of random Lipschitz functions on a lattice. The discussion introduces a diverse set of tools, useful for this purpose and for other problems, including spatial mixing, entropy and coupling methods, Gibbs measures and their classification and refined contour analysis.Comment: 53 pages, 10 figures; Based on lectures given at the workshop on Random Walks, Random Graphs and Random Media, September 2019, Munich and at the school Lectures on Probability and Stochastic Processes XIV, December 2019, Delh