Passage time from four to two blocks of opinions in the voter model and walks in the quarter plane

A random walk in $Z_+^2$ spatially homogeneous in the interior, absorbed at the axes, starting from an arbitrary point $(i_0,j_0)$ and with step probabilities drawn on Figure 1 is considered. The trivariate generating function of probabilities that the random walk hits a given point $(i,j)\in Z_+^2$ at a given time $k\geq 0$ is made explicit. Probabilities of absorption at a given time $k$ and at a given axis are found, and their precise asymptotic is derived as the time $k\to\infty$. The equivalence of two typical ways of conditioning this random walk to never reach the axes is established. The results are also applied to the analysis of the voter model with two candidates and initially, in the population $Z$, four connected blocks of same opinions. Then, a citizen changes his mind at a rate proportional to the number of its neighbors that disagree with him. Namely, the passage from four to two blocks of opinions is studied.Comment: 11 pages, 1 figur

On the functions counting walks with small steps in the quarter plane

Models of spatially homogeneous walks in the quarter plane ${\bf Z}_+^{2}$ with steps taken from a subset $\mathcal{S}$ of the set of jumps to the eight nearest neighbors are considered. The generating function $(x,y,z)\mapsto Q(x,y;z)$ of the numbers $q(i,j;n)$ of such walks starting at the origin and ending at $(i,j) \in {\bf Z}_+^{2}$ after $n$ steps is studied. For all non-singular models of walks, the functions $x \mapsto Q(x,0;z)$ and $y\mapsto Q(0,y;z)$ are continued as multi-valued functions on ${\bf C}$ having infinitely many meromorphic branches, of which the set of poles is identified. The nature of these functions is derived from this result: namely, for all the 51 walks which admit a certain infinite group of birational transformations of ${\bf C}^2$, the interval $]0,1/|\mathcal{S}|[$ of variation of $z$ splits into two dense subsets such that the functions $x \mapsto Q(x,0;z)$ and $y\mapsto Q(0,y;z)$ are shown to be holonomic for any $z$ from the one of them and non-holonomic for any $z$ from the other. This entails the non-holonomy of $(x,y,z)\mapsto Q(x,y;z)$, and therefore proves a conjecture of Bousquet-M\'elou and Mishna.Comment: 40 pages, 17 figure

Leptospira spp. strain identification by MALDI TOF MS is an equivalent tool to 16S rRNA gene sequencing and multi locus sequence typing (MLST)

Background: In this study mass spectrometry was used for evaluating extracted leptospiral protein samples and results were compared with molecular typing methods. For this, an extraction protocol for Leptospira spp. was independently established in two separate laboratories. Reference spectra were created with 28 leptospiral strains, including pathogenic, non-pathogenic and intermediate strains. This set of spectra was then evaluated on the basis of measurements with well-defined, cultured leptospiral strains and with 16 field isolates of veterinary or human origin. To verify discriminating peaks for the applied pathogenic strains, statistical analysis of the protein spectra was performed using the software tool ClinProTools. In addition, a dendrogram of the reference spectra was compared with phylogenetic trees of the 16S rRNA gene sequences and multi locus sequence typing (MLST) analysis. Results: Defined and reproducible protein spectra using MALDI-TOF MS were obtained for all leptospiral strains. Evaluation of the newly-built reference spectra database allowed reproducible identification at the species level for the defined leptospiral strains and the field isolates. Statistical analysis of three pathogenic genomospecies revealed peak differences at the species level and for certain serovars analyzed in this study. Specific peak patterns were reproducibly detected for the serovars Tarassovi, Saxkoebing, Pomona, Copenhageni, Australis, Icterohaemorrhagiae and Grippotyphosa. Analysis of the dendrograms of the MLST data, the 16S rRNA sequencing, and the MALDI-TOF MS reference spectra showed comparable clustering. Conclusions: MALDI-TOF MS analysis is a fast and reliable method for species identification, although Leptospira organisms need to be produced in a time-consuming culture process. All leptospiral strains were identified, at least at the species level, using our described extraction protocol. Statistical analysis of the three genomospecies L. borgpetersenii, L. interrogans and L. kirschneri revealed distinctive, reproducible differentiating peaks for seven leptospiral strains which represent seven serovars. Results obtained by MALDI-TOF MS were confirmed by MLST and 16S rRNA gene sequencing

Random walks reaching against all odds the other side of the quarter plane

For a homogeneous random walk in the quarter plane with nearest-neighbor transitions, starting from some state $(i_0,j_0)$, we study the event that the walk reaches the vertical axis, before reaching the horizontal axis. We derive an exact expression for the probability of this event, and derive an asymptotic expression for the case when $i_0$ becomes large, a situation in which the event becomes highly unlikely. The exact expression follows from the solution of a boundary value problem and is in terms of an integral that involves a conformal gluing function. The asymptotic expression follows from the asymptotic evaluation of this integral. Our results find applications in a model for nucleosome shifting, the voter model and the asymmetric exclusion process

A HUMAN PROOF OF GESSEL’S LATTICE PATH CONJECTURE (PRELIMINARY VERSION)

Abstract. Gessel walks are planar walks confined to the positive quarter plane, that move by unit steps in any of the following directions: West, North-East, East and South-West. In 2001, Ira Gessel conjectured a closed-form expression for the number of Gessel walks of a given length starting and ending at the origin. In 2008, Kauers, Koutschan and Zeilberger gave a computer-aided proof of this conjecture. The same year, Bostan and Kauers showed, using again computer algebra tools, that the trivariate generating function of Gessel walks is algebraic. In this article we propose the first “human proofs ” of these results. They are derived from a new expression for the generating function of Gessel walks. hal-00858083, version 1- 4 Sep 2013 1

3D positive lattice walks and spherical triangles

International audienceIn this paper we explore the asymptotic enumeration of three-dimensional excursions confined to the positive octant. As shown in [29], both the exponential growth and the critical exponent admit universal formulas, respectively in terms of the inventory of the step set and of the principal Dirichlet eigenvalue of a certain spherical triangle, itself being characterized by the steps of the model. We focus on the critical exponent, and our main objective is to relate combinatorial properties of the step set (structure of the so-called group of the walk, existence of a Hadamard factorization, existence of differential equations satisfied by the generating functions) to geometric or analytic properties of the associated spherical triangle (remarkable angles, tiling properties, existence of an exceptional closed-form formula for the principal eigenvalue). As in general the eigenvalues of the Dirichlet problem on a spherical triangle are not known in closed form, we also develop a finite-elements method to compute approximate values, typically with ten digits of precision

The compensation approach for walks with small steps in the quarter plane

This paper is the first application of the compensation approach to counting problems. We discuss how this method can be applied to a general class of walks in the quarter plane $Z_{+}^{2}$ with a step set that is a subset of $\{(-1,1),(-1,0),(-1,-1),(0,-1),(1,-1)\}$ in the interior of $Z_{+}^{2}$. We derive an explicit expression for the counting generating function, which turns out to be meromorphic and nonholonomic, can be easily inverted, and can be used to obtain asymptotic expressions for the counting coefficients

ON THE NATURE OF THE GENERATING SERIES OF WALKS IN THE QUARTER PLANE

International audienceIn the present paper, we introduce a new approach, relying on the Galois theory of difference equations, to study the nature of the generating series of walks in the quarter plane. Using this approach, we are not only able to recover many of the recent results about these series, but also to go beyond them. For instance, we give for the first time hypertranscendency results, i.e., we prove that certain of these generating series do not satisfy any nontrivial nonlinear algebraic differential equation with rational coefficients