Fully Analyzing an Algebraic Polya Urn Model

This paper introduces and analyzes a particular class of Polya urns: balls are of two colors, can only be added (the urns are said to be additive) and at every step the same constant number of balls is added, thus only the color compositions varies (the urns are said to be balanced). These properties make this class of urns ideally suited for analysis from an "analytic combinatorics" point-of-view, following in the footsteps of Flajolet-Dumas-Puyhaubert, 2006. Through an algebraic generating function to which we apply a multiple coalescing saddle-point method, we are able to give precise asymptotic results for the probability distribution of the composition of the urn, as well as local limit law and large deviation bounds.Comment: LATIN 2012, Arequipa : Peru (2012

Laguerre-type derivatives: Dobinski relations and combinatorial identities

We consider properties of the operators D(r,M)=a^r(a^\dag a)^M (which we call generalized Laguerre-type derivatives), with r=1,2,..., M=0,1,..., where a and a^\dag are boson annihilation and creation operators respectively, satisfying [a,a^\dag]=1. We obtain explicit formulas for the normally ordered form of arbitrary Taylor-expandable functions of D(r,M) with the help of an operator relation which generalizes the Dobinski formula. Coherent state expectation values of certain operator functions of D(r,M) turn out to be generating functions of combinatorial numbers. In many cases the corresponding combinatorial structures can be explicitly identified.Comment: 14 pages, 1 figur

Phase Transition in the Aldous-Shields Model of Growing Trees

We study analytically the late time statistics of the number of particles in a growing tree model introduced by Aldous and Shields. In this model, a cluster grows in continuous time on a binary Cayley tree, starting from the root, by absorbing new particles at the empty perimeter sites at a rate proportional to c^{-l} where c is a positive parameter and l is the distance of the perimeter site from the root. For c=1, this model corresponds to random binary search trees and for c=2 it corresponds to digital search trees in computer science. By introducing a backward Fokker-Planck approach, we calculate the mean and the variance of the number of particles at large times and show that the variance undergoes a `phase transition' at a critical value c=sqrt{2}. While for c>sqrt{2} the variance is proportional to the mean and the distribution is normal, for c<sqrt{2} the variance is anomalously large and the distribution is non-Gaussian due to the appearance of extreme fluctuations. The model is generalized to one where growth occurs on a tree with $m$ branches and, in this more general case, we show that the critical point occurs at c=sqrt{m}.Comment: Latex 17 pages, 6 figure

Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces

The Airy distribution function describes the probability distribution of the area under a Brownian excursion over a unit interval. Surprisingly, this function has appeared in a number of seemingly unrelated problems, mostly in computer science and graph theory. In this paper, we show that this distribution also appears in a rather well studied physical system, namely the fluctuating interfaces. We present an exact solution for the distribution P(h_m,L) of the maximal height h_m (measured with respect to the average spatial height) in the steady state of a fluctuating interface in a one dimensional system of size L with both periodic and free boundary conditions. For the periodic case, we show that P(h_m,L)=L^{-1/2}f(h_m L^{-1/2}) for all L where the function f(x) is the Airy distribution function. This result is valid for both the Edwards-Wilkinson and the Kardar-Parisi-Zhang interfaces. For the free boundary case, the same scaling holds P(h_m,L)=L^{-1/2}F(h_m L^{-1/2}), but the scaling function F(x) is different from that of the periodic case. We compute this scaling function explicitly for the Edwards-Wilkinson interface and call it the F-Airy distribution function. Numerical simulations are in excellent agreement with our analytical results. Our results provide a rather rare exactly solvable case for the distribution of extremum of a set of strongly correlated random variables. Some of these results were announced in a recent Letter [ S.N. Majumdar and A. Comtet, Phys. Rev. Lett., 92, 225501 (2004)].Comment: 27 pages, 10 .eps figures included. Two figures improved, new discussion and references adde

Norbin ablation results in defective adult hippocampal neurogenesis and depressive-like behavior in mice

Adult neurogenesis in the hippocampus subgranular zone is associated with the etiology and treatment efficiency of depression. Factors that affect adult hippocampal neurogenesis have been shown to contribute to the neuropathology of depression. Glutamate, the major excitatory neurotransmitter, plays a critical role in different aspects of neurogenesis. Of the eight metabotropic glutamate receptors (mGluRs), mGluR5 is the most highly expressed in neural stem cells. We previously identified Norbin as a positive regulator of mGluR5 and showed that its expression promotes neurite outgrowth. In this study, we investigated the role of Norbin in adult neurogenesis and depressive-like behaviors using Norbin-deficient mice. We found that Norbin deletion significantly reduced hippocampal neurogenesis; specifically, the loss of Norbin impaired the proliferation and maturation of newborn neurons without affecting cell-fate specification of neural stem cells/neural progenitor cells (NSCs/NPCs). Norbin is highly expressed in the granular neurons in the dentate gyrus of the hippocampus, but it is undetectable in NSCs/NPCs or immature neurons, suggesting that the effect of Norbin on neurogenesis is likely caused by a nonautonomous niche effect. In support of this hypothesis, we found that the expression of a cell-cell contact gene, Desmoplakin, is greatly reduced in Norbin-deletion mice. Moreover, Norbin-KO mice show an increased immobility in the forced-swim test and the tail-suspension test and reduced sucrose preference compared with wild-type controls. Taken together, these results show that Norbin is a regulator of adult hippocampal neurogenesis and that its deletion causes depressive-like behaviors

Understanding Search Trees via Statistical Physics

We study the random m-ary search tree model (where m stands for the number of branches of a search tree), an important problem for data storage in computer science, using a variety of statistical physics techniques that allow us to obtain exact asymptotic results. In particular, we show that the probability distributions of extreme observables associated with a random search tree such as the height and the balanced height of a tree have a traveling front structure. In addition, the variance of the number of nodes needed to store a data string of a given size N is shown to undergo a striking phase transition at a critical value of the branching ratio m_c=26. We identify the mechanism of this phase transition, show that it is generic and occurs in various other problems as well. New results are obtained when each element of the data string is a D-dimensional vector. We show that this problem also has a phase transition at a critical dimension, D_c= \pi/\sin^{-1}(1/\sqrt{8})=8.69363...Comment: 11 pages, 8 .eps figures included. Invited contribution to STATPHYS-22 held at Bangalore (India) in July 2004. To appear in the proceedings of STATPHYS-2

On Ramanujan's Q-function

This study provides a detailed analysis of a function which Knuth discovered to play a central role in the analysis of hashing with linear probing. The function, named after Knuth Q(n), is related to several of Ramanujan's investigations. It surfaces in the analysis of a variety of algorithms ans discrete probability problems including hashing, the birthday paradox, random mapping statistics, the "rho" method for integer factorization, union-find algorithms, optimum caching, and the study of memory conflicts. A process related to the complex asymptotic methods of singularity analysis and saddle point integrals permits to precisely quantify the behaviour of the Q(n) function. in this way, tight bounds are derived. They answer a question of Knuth (the art of Computer Programming, vol. 1, 1968), itself a rephrasing of earlier questions of Ramanujan in 1911-1913

Counting, generating and sampling tree alignments

Pairwise ordered tree alignment are combinatorial objects that appear in RNA secondary structure comparison. However, the usual representation of tree alignments as supertrees is ambiguous, i.e. two distinct supertrees may induce identical sets of matches between identical pairs of trees. This ambiguity is uninformative, and detrimental to any probabilistic analysis.In this work, we consider tree alignments up to equivalence. Our first result is a precise asymptotic enumeration of tree alignments, obtained from a context-free grammar by mean of basic analytic combinatorics. Our second result focuses on alignments between two given ordered trees $S$ and $T$. By refining our grammar to align specific trees, we obtain a decomposition scheme for the space of alignments, and use it to design an efficient dynamic programming algorithm for sampling alignments under the Gibbs-Boltzmann probability distribution. This generalizes existing tree alignment algorithms, and opens the door for a probabilistic analysis of the space of suboptimal RNA secondary structures alignments.Comment: ALCOB - 3rd International Conference on Algorithms for Computational Biology - 2016, Jun 2016, Trujillo, Spain. 201

Heisenberg-Weyl algebra revisited: Combinatorics of words and paths

The Heisenberg-Weyl algebra, which underlies virtually all physical representations of Quantum Theory, is considered from the combinatorial point of view. We provide a concrete model of the algebra in terms of paths on a lattice with some decomposition rules. We also discuss the rook problem on the associated Ferrers board; this is related to the calculus in the normally ordered basis. From this starting point we explore a combinatorial underpinning of the Heisenberg-Weyl algebra, which offers novel perspectives, methods and applications.Comment: 5 pages, 3 figure