Motzkin paths, Motzkin polynomials and recurrence relations

We consider the Motzkin paths which are simple combinatorial objects appearing in many contexts. They are counted by the Motzkin numbers, related to the well known Catalan numbers. Associated with the Motzkin paths, we introduce the Motzkin polynomial, which is a multi-variable polynomial "counting" all Motzkin paths of a certain type. Motzkin polynomials (also called Jacobi-Rogers polynomials) have been studied before, but here we deduce sonic properties based on recurrence relations. The recurrence relations proved here also allow an efficient computation of the Motzkin polynomials. Finally, we show that the matrix entries of powers of an arbitrary tridiagonal matrix are essentially given by Motzkin polynomials, a property commonly known but usually stated without proof

Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Asymptotic Aspects and Borges\u27s Theorem

In a companion article dedicated to the enumeration aspects, we showed how to obtain closed form formulas for the generating functions of walks, bridges, meanders, and excursions avoiding any fixed word (a pattern p). The autocorrelation polynomial of this forbidden pattern p (as introduced by Guibas and Odlyzko in 1981, in the context of regular expressions) plays a crucial role. In this article, we get the asymptotics of these walks. We also introduce a trivariate generating function (length, final altitude, number of occurrences of p), for which we derive a closed form. We prove that the number of occurrences of p is normally distributed: This is what Flajolet and Sedgewick call an instance of Borges\u27s theorem. We thus extend and refine the study by Banderier and Flajolet in 2002 on lattice paths, and we unify several dozens of articles which investigated patterns like peaks, valleys, humps, etc., in Dyck and Motzkin paths. Our approach relies on methods of analytic combinatorics, and on a matricial generalization of the kernel method. The situation is much more involved than in the Banderier-Flajolet work: forbidden patterns lead to a wider zoology of asymptotic behaviours, and we classify them according to the geometry of a Newton polygon associated with these constrained walks, and we analyse what are the universal phenomena common to all these models of lattice paths avoiding a pattern

Distribution of peak heights modulo kk and double descents on kk-Dyck paths

We show that the distribution of the number of peaks at height $i$ modulo $k$ in $k$-Dyck paths of a given length is independent of $i\in[0,k-1]$ and is the reversal of the distribution of the total number of peaks. Moreover, these statistics, together with the number of double descents, are jointly equidistributed with any of their permutations. We also generalize this result to generalized Motzkin paths and generalized ballot paths.Comment: 11 pages, 3 figure

On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution

International audienc

Nonlinear Lattices and Random Matrix

In this thesis, we study problems related to statistical properties of integrable and non-integrable Hamiltonian system, focusing on their relations with random matrix theory. First, we consider the harmonic chain with short-range interactions. Exploiting the rich theory of circulant and Toeplitz matrices, we are able to explicitly compute the correlation functions for this system. Further, applying the so-called steepest descent method, we compute their long time asymptotic. In the main part of the thesis, we focus on the interplay between Random Matrix theory and integrable Hamiltonian system. Specifically, we introduce some new tridiagonal random matrix ensembles that we name $alpha$ ensembles, and we compute their mean density of states. These random matrix models are related to the classical beta ones in the high temperature regime. Moreover, they are also connected to the Toda and the Ablowitz-Ladik lattice, indeed applying our result on the $alpha$ ensembles, we are able to compute the mean density of states of the Lax matrices of these two lattices. Next, we focus on the Fermi-Pasta-Ulam-Tsingou (FPUT) system, a non-integrable lattice. We show that the integrals of motion of the Toda lattice are adiabatic invariants, namely statistically almost conserved quantities, for the FPUT system for a time-scale of order $eta^{1-2arepsilon}$, here arepsilon>0, and $eta$ is the inverse of the temperature. Moreover, we show that some special linear combinations of the normal modes are adiabatic invariants for the Toda lattice, for all times, and for the FPUT, for times of order $eta^{1-2arepsilon}$. Finally, we consider the classical beta ensembles in the high temperature regime. We compute their mean density of states, making use of the so-called loop equations. Exploiting this formalism, we are able to compute the moments and the linear statistic covariance through recurrence relations. Further, we identify a new $alpha$ ensemble, which is related to Dyson's study of a disordered chain. Our analysis supplement the results contained in Dyson's work