We study scaling limits of a family of planar random growth processes in which clusters grow by the successive aggregation of small particles. In these models, clusters are encoded as a composition of conformal maps and the attachment point of each successive particle is distributed according to harmonic measure on the cluster boundary, raised to some power. We show that, when this power lies within a particular range, the macroscopic shape of the cluster converges to a disk, but that as the power approaches the edge of this range the fluctuations approach a critical point, which is a limit of stability.
In @cite_13 , Sola, Turner and Viklund showed the existence of a phase transition in the ALE( @math ) model at @math by focussing on the regime @math . They showed, in this case, that if particles are taken to be slits, and the regularisation parameter @math is sufficiently small, then the clusters converge to macroscopic growing slits. Almost all other previous work relates to HL(0). In @cite_1 , Norris and Turner showed that the HL(0) cluster converges to a disk with internal branching structure given by the Brownian web. More recently, Silvestri @cite_9 analysed the fluctuations in HL(0) and showed that these converge to a log-correlated fractional Gaussian field. Several other papers consider modifications of the HL(0) model @cite_2 @cite_11 @cite_8 .
@cite_8 @cite_9 @cite_1 @cite_2 @cite_13 @cite_11
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We study a regularized version of Hastings-Levitov planar random growth that models clusters formed by the aggregation of diffusing particles. In this model, the growing clusters are defined in terms of iterated slit maps whose capacities are given by c_n=c| n-1 '(e^ +i )|^ - , 0, where c>0 is the capacity of the first particle, _n are the composed conformal maps defining the clusters of the evolution, _n are independent uniform angles determining the positions at which particles are attached, and >0 is a regularization parameter which we take to depend on c. We prove that under an appropriate rescaling of time, in the limit as c converges to 0, the clusters converge to growing disks with deterministic capacities, provided that does not converge to 0 too fast. We then establish scaling limits for the harmonic measure flow, showing that by letting tend to 0 at different rates it converges to either the Brownian web on the circle, a stopped version of the Brownian web on the circle, or the identity map. As the harmonic measure flow is closely related to the internal branching structure within the cluster, the above three cases intuitively correspond to the number of infinite branches in the model being either 1, a random number whose distribution we obtain, or unbounded, in the limit as c converges to 0. We also present several findings based on simulations of the model with parameter choices not covered by our rigorous analysis. We study an evolving spatial network in which sequentially arriving vertices are joined to existing vertices at random according to a rule that combines preference according to degree with preference according to spatial proximity. We investigate phase transitions in graph structure as the relative weighting of these two components of the attachment rule is varied. Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence of the resulting graph coincides with that of the standard Barab 'asi--Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence coincides with that of a purely geometric model, the on-line nearest-neighbour graph, which is of interest in its own right and for which we prove some extensions of known results. We also show the presence of an intermediate regime, in which the behaviour differs significantly from both the on-line nearest-neighbour graph and the Barab 'asi--Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence. Our results lend some mathematical support to simulation studies of Manna and Sen, while proving that the power law to stretched exponential phase transition occurs at a different point from the one conjectured by those authors. New phase transition phenomena have recently been discovered for the stochastic block model, for the special case of two non-overlapping symmetric communities. This gives raise in particular to new algorithmic challenges driven by the thresholds. This paper investigates whether a general phenomenon takes place for multiple communities, without imposing symmetry. In the general stochastic block model @math , @math vertices are split into @math communities of relative size @math , and vertices in community @math and @math connect independently with probability @math . This paper investigates the partial and exact recovery of communities in the general SBM (in the constant and logarithmic degree regimes), and uses the generality of the results to tackle overlapping communities. The contributions of the paper are: (i) an explicit characterization of the recovery threshold in the general SBM in terms of a new divergence function @math , which generalizes the Hellinger and Chernoff divergences, and which provides an operational meaning to a divergence function analog to the KL-divergence in the channel coding theorem, (ii) the development of an algorithm that recovers the communities all the way down to the optimal threshold and runs in quasi-linear time, showing that exact recovery has no information-theoretic to computational gap for multiple communities, in contrast to the conjectures made for detection with more than 4 communities; note that the algorithm is optimal both in terms of achieving the threshold and in having quasi-linear complexity, (iii) the development of an efficient algorithm that detects communities in the constant degree regime with an explicit accuracy bound that can be made arbitrarily close to 1 when a prescribed signal-to-noise ratio (defined in term of the spectrum of @math ) tends to infinity. A branching random walk in presence of an absorbing wall moving at a constant velocity v undergoes a phase transition as v varies. The problem can be analyzed using the properties of the Fisher-Kolmogorov-Petrovsky-Piscounov (F-KPP) equation. We find that the survival probability of the branching random walk vanishes at a critical velocity vc of the wall with an essential singularity and we characterize the divergences of the relaxation times for v vc. At v=vc the survival probability decays like a stretched exponential. Using the F-KPP equation, one can also calculate the distribution of the population size at time t conditioned by the survival of one individual at a later time T>t. Our numerical results indicate that the size of the population diverges like the exponential of (vc−v)−1 2 in the quasi-stationary regime below vc. Moreover for v>vc, our data indicate that there is no quasi-stationary regime.