Gregory Schehr

Collective dynamics of social annotation

The enormous increase of popularity and use of the worldwide web has led in the recent years to important changes in the ways people communicate. An interesting example of this fact is provided by the now very popular social annotation systems, through which users annotate resources (such as web pages or digital photographs) with keywords known as “tags.” Understanding the rich emergent structures resulting from the uncoordinated actions of users calls for an interdisciplinary effort. In particular concepts borrowed from statistical physics, such as random walks (RWs), and complex networks theory, can effectively contribute to the mathematical modeling of social annotation systems. Here, we show that the process of social annotation can be seen as a collective but uncoordinated exploration of an underlying semantic space, pictured as a graph, through a series of RWs. This modeling framework reproduces several aspects, thus far unexplained, of social annotation, among which are the peculiar growth of the size of the vocabulary used by the community and its complex network structure that represents an externalization of semantic structures grounded in cognition and that are typically hard to access.

Non-intersecting Brownian walkers and Yang–Mills theory on the sphere

We study a system of N non-intersecting Brownian motions on a line segment [0,L] with periodic, absorbing and reflecting boundary conditions. We show that the normalized reunion probabilities of these Brownian motions in the three models can be mapped to the partition function of two-dimensional continuum Yang–Mills theory on a sphere respectively with gauge groups U(N), Sp(2N) and SO(2N). Consequently, we show that in each of these Brownian motion models, as one varies the system size L, a third order phase transition occurs at a critical value L=Lc(N)∼N in the large N limit. Close to the critical point, the reunion probability, properly centered and scaled, is identical to the Tracy–Widom distribution describing the probability distribution of the largest eigenvalue of a random matrix. For the periodic case we obtain the Tracy–Widom distribution corresponding to the GUE random matrices, while for the absorbing and reflecting cases we get the Tracy–Widom distribution corresponding to GOE random matrices. In the absorbing case, the reunion probability is also identified as the maximal height of N non-intersecting Brownian excursions (“watermelons” with a wall) whose distribution in the asymptotic scaling limit is then described by GOE Tracy–Widom law. In addition, large deviation formulas for the maximum height are also computed.

Top eigenvalue of a random matrix: large deviations and third order phase transition

We study the fluctuations of the largest eigenvalue ?max of N ? N random matrices in the limit of large N. The main focus is on Gaussian ? ensembles, including in particular the Gaussian orthogonal (? = 1), unitary (? = 2) and symplectic (? = 4) ensembles. The probability density function (PDF) of ?max consists, for large N, of a central part described by Tracy?Widom distributions flanked, on both sides, by two large deviation tails. While the central part characterizes the typical fluctuations of ?max?of order ?the large deviation tails are instead associated with extremely rare fluctuations?of order . Here we review some recent developments in the theory of these extremely rare events using a Coulomb gas approach. We discuss in particular the third order phase transition which separates the left tail from the right tail, a transition akin to the so-called Gross?Witten?Wadia phase transition found in 2-d lattice quantum chromodynamics. We also discuss the occurrence of similar third order transitions in various physical problems, including non-intersecting Brownian motions, conductance fluctuations in mesoscopic physics and entanglement in a bipartite system.

A note on q-Gaussians and non-Gaussians in statistical mechanics

The sum of N sufficiently strongly correlated random variables will not in general be Gaussian distributed in the limit . We revisit examples of sums x that have recently been put forward as instances of variables obeying a q-Gaussian law, that is, one of type cst × [1−(1−q)x2]1/(1−q). We show by explicit calculation that the probability distributions in the examples are actually analytically different from q-Gaussians, in spite of numerically resembling them very closely. Although q-Gaussians exhibit many interesting properties, the examples investigated do not support the idea that they play a special role as limit distributions of correlated sums.

Exact distribution of the maximal height of p vicious walkers

Using path-integral techniques, we compute exactly the distribution of the maximal height Hp of p nonintersecting Brownian walkers over a unit time interval in one dimension, both for excursions p watermelons with a wall, and bridges p watermelons without a wall, for all integer p>or=1. For large p, we show that approximately square root 2p (excursions) whereas approximately square root p (bridges). Our exact results prove that previous numerical experiments only measured the preasymptotic behaviors and not the correct asymptotic ones. In addition, our method establishes a physical connection between vicious walkers and random matrix theory.

First passages for a search by a swarm of independent random searchers

In this paper we study some aspects of search for an immobile target by a swarm of N non-communicating, randomly moving searchers (numbered by the index k, k = 1, 2,..., N), which all start their random motion simultaneously at the same point in space. For each realization of the search process, we record the unordered set of time moments {?k}, where ?k is the time of the first passage of the kth searcher to the location of the target. Clearly, ?ks are independent, identically distributed random variables with the same distribution function ?(?). We evaluate then the distribution P(?) of the random variable , where is the ensemble-averaged realization-dependent first passage time. We show that P(?) exhibits quite a non-trivial and sometimes a counterintuitive behavior. We demonstrate that in some well-studied cases (e.g. Brownian motion in finite d-dimensional domains) the mean first passage time is not a robust measure of the search efficiency, despite the fact that ?(?) has moments of arbitrary order. This implies, in particular, that even in this simplest case (not to mention complex systems and/or anomalous diffusion) first passage data extracted from a single-particle tracking should be regarded with appropriate caution because of the significant sample-to-sample fluctuations.

Extremes of N Vicious Walkers for Large N: Application to the Directed Polymer and KPZ Interfaces

We compute the joint probability density function (jpdf) PN(M,τM) of the maximum M and its position τM for N non-intersecting Brownian excursions, on the unit time interval, in the large N limit. For N→∞, this jpdf is peaked around

Extreme value statistics from the real space renormalization group: Brownian motion, Bessel processes and continuous time random walks

M = \sqrt{2N}

On the joint distribution of the maximum and its position of the Airy2 process minus a parabola

and τM=1/2, while the typical fluctuations behave for large N like

Extremal statistics of curved growing interfaces in 1+1 dimensions

M - \sqrt{2N} \propto s N^{-1/6}