Abstract:
We study the density of clustered immobile reactants in the diffusion-controlled single species annihilation. An initial state in which these impurities occupy a subspace of codimension d' leads to a substantial enhancement of their survival probability. The Smoluchowski rate theory suggests that the codimensionality plays a crucial role in determining the long time behavior. The system undergoes a transition at d'=2. For d'<2, a finite fraction of the impurities survive: ni(t) ~ ni(infinity)+const x log(t)/t^{1/2} for d=2 and ni(t) ~ ni(infinity)+const/t^{1/2} for d>2. Above this critical codimension, d'>=2, the subspace decays indefinitely. At the critical codimension, inverse logarithmic decay occurs, ni(t) ~ log(t)^{-a(d,d')}. Above the critical codimension, the decay is algebraic ni(t) ~ t^{-a(d,d')}. In general, the exponents governing the long time behavior depend on the dimension as well as the codimension.

Abstract:
We analyze the evolution of political organizations using a model in which agents change their opinions via two competing mechanisms. Two agents may interact and reach consensus, and additionally, individual agents may spontaneously change their opinions by a random, diffusive process. We find three distinct possibilities. For strong diffusion, the distribution of opinions is uniform and no political organizations (parties) are formed. For weak diffusion, parties do form and furthermore, the political landscape continually evolves as small parties merge into larger ones. Without diffusion, a pattern develops: parties have the same size and they possess equal niches. These phenomena are analyzed using pattern formation and scaling techniques.

Abstract:
We study how the order of N independent random walks in one dimension evolves with time. Our focus is statistical properties of the inversion number m, defined as the number of pairs that are out of sort with respect to the initial configuration. In the steady-state, the distribution of the inversion number is Gaussian with the average ~N^2/4 and the standard deviation sigma N^{3/2}/6. The survival probability, S_m(t), which measures the likelihood that the inversion number remains below m until time t, decays algebraically in the long-time limit, S_m t^{-beta_m}. Interestingly, there is a spectrum of N(N-1)/2 distinct exponents beta_m(N). We also find that the kinetics of first-passage in a circular cone provides a good approximation for these exponents. When N is large, the first-passage exponents are a universal function of a single scaling variable, beta_m(N)--> beta(z) with z=(m-)/sigma. In the cone approximation, the scaling function is a root of a transcendental equation involving the parabolic cylinder equation, D_{2 beta}(-z)=0, and surprisingly, numerical simulations show this prediction to be exact.

Abstract:
We study structural properties of trees grown by preferential attachment. In this mechanism, nodes are added sequentially and attached to existing nodes at a rate that is strictly proportional to the degree. We classify nodes by their depth n, defined as the distance from the root of the tree, and find that the network is strongly stratified. Most notably, the distribution f_k^(n) of nodes with degree k at depth n has a power-law tail, f_k^(n) ~ k^{-\gamma(n)}. The exponent grows linearly with depth, gamma(n)=2+(n-1)/, where the brackets denote an average over all nodes. Therefore, nodes that are closer to the root are better connected, and moreover, the degree distribution strongly varies with depth. Similarly, the in-component size distribution has a power-law tail and the characteristic exponent grows linearly with depth. Qualitatively, these behaviors extend to a class of networks that grow by a redirection mechanism.

Abstract:
We study statistical properties of the Kolmogorov-Avrami-Johnson-Mehl nucleation-and-growth model in one dimension. We obtain exact results for the gap density as well as the island distribution. When all nucleation events occur simultaneously, the island distribution has discontinuous derivatives on the rays x_n(t)=nt, n=1,2,3... We introduce an accelerated growth mechanism where the velocity increases linearly with the island size. We solve for the inter-island gap density and show that the system reaches complete coverage in a finite time and that the near-critical behavior of the system is robust, i.e., it is insensitive to details such as the nucleation mechanism.

Abstract:
We study transport of interacting particles in weakly disordered media. Our one-dimensional system includes (i) disorder: the hopping rate governing the movement of a particle between two neighboring lattice sites is inhomogeneous, and (ii) hard core interaction: the maximum occupancy at each site is one particle. We find that over a substantial regime, the root-mean-square displacement of a particle, sigma, grows super-diffusively with time t, sigma ~ (epsilon t)^{2/3}, where epsilon is the disorder strength. Without disorder the particle displacement is sub-diffusive, sigma ~ t^{1/4}, and therefore disorder dramatically enhances particle mobility. We explain this effect using scaling arguments, and verify the theoretical predictions through numerical simulations. Also, the simulations show that disorder generally leads to stronger mobility.

Abstract:
We investigate a network growth model in which the genealogy controls the evolution. In this model, a new node selects a random target node and links either to this target node, or to its parent, or to its grandparent, etc; all nodes from the target node to its most ancient ancestor are equiprobable destinations. The emerging random ancestor tree is very shallow: the fraction g_n of nodes at distance n from the root decreases super-exponentially with n, g_n=e^{-1}/(n-1)!. We find that a macroscopic hub at the root coexists with highly connected nodes at higher generations. The maximal degree of a node at the nth generation grows algebraically as N^{1/beta_n} where N is the system size. We obtain the series of nontrivial exponents which are roots of transcendental equations: beta_1= 1.351746, beta_2=1.682201, etc. As a consequence, the fraction p_k of nodes with degree k has algebraic tail, p_k ~ k^{-gamma}, with gamma=beta_1+1=2.351746.

Abstract:
We study dynamical ordering of rods. In this process, rod alignment via pairwise interactions competes with diffusive wiggling. Under strong diffusion, the system is disordered, but at weak diffusion, the system is ordered. We present an exact steady-state solution for the nonlinear and nonlocal kinetic theory of this process. We find the Fourier transform as a function of the order parameter, and show that Fourier modes decay exponentially with the wave number. We also obtain the order parameter in terms of the diffusion constant. This solution is obtained using iterated partitions of the integer numbers.

Abstract:
Aggregation processes with an arbitrary number of conserved quantities are investigated. On the mean-field level, an exact solution for the size distribution is obtained. The asymptotic form of this solution exhibits nontrivial ``double'' scaling. While processes with one conserved quantity are governed by a single scale, processes with multiple conservation laws exhibit an additional diffusion-like scale. The theory is applied to ballistic aggregation with mass and momentum conserving collisions and to diffusive aggregation with multiple species.

Abstract:
Irreversible aggregation processes involving reactive and frozen clusters are investigated using the rate equation approach. In aggregation events, two clusters join irreversibly to form a larger cluster, and additionally, reactive clusters may spontaneously freeze. Frozen clusters do not participate in merger events. Generally, freezing controls the nature of the aggregation process, as demonstrated by the final distribution of frozen clusters. The cluster mass distribution has a power-law tail, F_k ~ k^{-gamma}, when the freezing process is sufficiently slow. Different exponents, gamma=1, 3 are found for the constant and the product aggregation rates, respectively. For the latter case, the standard polymerization model, either no gels, or a single gel, or even multiple gels may be produced.