[en] The organizational development of growing random networks is investigated. These growing networks are built by adding nodes successively, and linking each to an earlier node of degree k with an attachment probability A_k. When A_k grows more slowly than linearly with k, the number of nodes with k links, N_k(t), decays faster than a power law in k, while for A_k growing faster than linearly in k, a single node emerges which connects to nearly all other nodes. When A_k is asymptotically linear, N_k(t)∼tk^-ν, with ν dependent on details of the attachment probability, but in the range 2<ν<∞. The combined age and degree distribution of nodes shows that old nodes typically have a large degree. There is also a significant correlation in the degrees of neighboring nodes, so that nodes of similar degree are more likely to be connected. The size distributions of the in and out components of the network with respect to a given node-namely, its 'descendants' and 'ancestors'-are also determined. The in component exhibits a robust s^-2 power-law tail, where s is the component size. The out component has a typical size of order lnt, and it provides basic insights into the genealogy of the network

[en] We investigate extremal statistical properties such as the maximal and the minimal heights of randomly generated binary trees. By analyzing the master evolution equations we show that the cumulative distribution of extremal heights approaches a traveling wave form. The wave front in the minimal case is governed by the small-extremal-height tail of the distribution, and conversely, the front in the maximal case is governed by the large-extremal-height tail of the distribution. We determine several statistical characteristics of the extremal height distribution analytically. In particular, the expected minimal and maximal heights grow logarithmically with the tree size, N, h_min∼v_min ln N, and h_max∼v_max ln N, with v_min=0.373365(hor ellipsis) and v_max=4.31107(hor ellipsis), respectively. Corrections to this asymptotic behavior are of order O(ln ln N)

[en] The in-degree and out-degree distributions of a growing network model are determined. The in-degree is the number of incoming links to a given node (and vice versa for out-degree). The network is built by (i) creation of new nodes which each immediately attach to a preexisting node, and (ii) creation of new links between preexisting nodes. This process naturally generates correlated in-degree and out-degree distributions. When the node and link creation rates are linear functions of node degree, these distributions exhibit distinct power-law forms. By tuning the parameters in these rates to reasonable values, exponents which agree with those of the web graph are obtained

[en] A ferromagnetic Ising chain which is endowed with a single-spin-flip Glauber dynamics is investigated. For an arbitrary annealing protocol, we derive an exact integral equation for the domain wall density. This integral equation admits an asymptotic solution in the limit of extremely slow cooling. For instance, we extract an asymptotic of the density of domain walls at the end of the cooling procedure when the temperature vanishes. Slow annealing is usually studied using a Kibble–Zurek argument; in our setting, this argument leads to approximate predictions which are in good agreement with exact asymptotics

[en] We investigate a system of interacting clusters evolving through mass exchange and supplemented by input of small clusters. Three possibilities depending on the rate of exchange generically occur when input is homogeneous: continuous growth, gelation, and instantaneous gelation. We mostly study the growth regime using scaling methods. An exchange process with reaction rates equal to the product of reactant masses admits an exact solution which allows us to justify the validity of scaling approaches in this special case. We also investigate exchange processes with a localized input. We show that if the diffusion coefficients are mass-independent, the cluster mass distribution becomes stationary and develops an algebraic tail far away from the source. (paper)

[en] We study an infinite system of particles initially occupying a half-line y ⩽ 0 and undergoing random walks on the entire line. The right-most particle is called a leader. Surprisingly, every particle except the original leader may never achieve the leadership throughout the evolution. For the equidistant initial configuration, the kth particle attains the leadership with probability e⁻² k ⁻¹(ln k)^−1/2 when k ≫ 1. This decay law provides a quantitative measure of the correlation between earlier misfortune proportional to the label k and eternal failure. We also show that the winner defined as the first walker overtaking the initial leader has label k ≫ 1 with probability decaying as $\mathrm{e}\mathrm{x}\mathrm{p}[-<!--\; ???\; -->\frac{1}{2}{\left(\mathrm{l}\mathrm{n}k\right)}^2]$. (paper)

[en] We study dynamical behaviors of one-dimensional stochastic lattice gases with repulsive interactions whose span can be arbitrarily large. We endow the system with a zero-temperature dynamics, so that hops to empty sites which would have led to an increase of energy are forbidden. We assume that the strength of interactions decreases sufficiently quickly with the separation between the particles, so that interactions can be treated in a lexicographic order. For such repulsion processes with symmetric nearest-neighbor hopping we analytically determine the density-dependent diffusion coefficient. We also compute the variance of the displacement of a tagged particle. (paper)

[en] We study the zero-temperature Ising chain evolving according to the Swendsen-Wang dynamics. We determine analytically the domain length distribution and various 'historical' characteristics, e.g. the density of unreacted domains is shown to scale with the average domain length as (l)^-δ with δ = 3/2 (for the q-state Potts model, δ = 1 + q^-1). We also compute the domain length distribution for the Ising chain endowed with the zero-temperature Wolff dynamics

[en] We investigate how the initial number of infected individuals affects the behavior of the critical susceptible-infected-recovered process. We analyze the outbreak size distribution, duration of the outbreaks, and the role of fluctuations. (paper)

[en] We show that a two-dimensional convection-diffusion problem with a radial sink or source at the origin may be recast as a pure diffusion problem in a fictitious space in which the spatial dimension is continuously tunable with the Peclet number. This formulation allows us to probe various diffusion-controlled processes in non-integer dimensions