Statistics of Superior Records

We study statistics of records in a sequence of random variables. These identical and independently distributed variables are drawn from the parent distribution rho. The running record equals the maximum of all elements in the sequence up to a given point. We define a superior sequence as one where all running records are above the average record, expected for the parent distribution rho. We find that the fraction of superior sequences S_N decays algebraically with sequence length N, S_N ~ N^{-beta} in the limit N-->infty. Interestingly, the decay exponent beta is nontrivial, being the root of an integral equation. For example, when rho is a uniform distribution with compact support, we find beta=0.450265. In general, the tail of the parent distribution governs the exponent beta. We also consider the dual problem of inferior sequences, where all records are below average, and find that the fraction of inferior sequences I_N decays algebraically, albeit with a different decay exponent, I_N ~ N^{-alpha}. We use the above statistical measures to analyze earthquake data.Comment: 8 pages, 6 figures, expanded versio

Scaling Exponent for Incremental Records

We investigate records in a growing sequence of identical and independently distributed random variables. The record equals the largest value in the sequence, and our focus is on the increment, defined as the difference between two successive records. We investigate sequences in which all increments decrease monotonically, and find that the fraction I_N of sequences that exhibit this property decays algebraically with sequence length N, namely I_N ~ N^{-nu} as N --> infinity. We analyze the case where the random variables are drawn from a uniform distribution with compact support, and obtain the exponent nu = 0.317621... using analytic methods. We also study the record distribution and the increment distribution. Whereas the former is a narrow distribution with an exponential tail, the latter is broad and has a power-law tail characterized by the exponent nu. Empirical analysis of records in the sequence of waiting times between successive earthquakes is consistent with the theoretical results.Comment: 7 pages, 8 figure

Kinetics of Ring Formation

We study reversible polymerization of rings. In this stochastic process, two monomers bond and as a consequence, two disjoint rings may merge into a compound ring, or, a single ring may split into two fragment rings. This aggregation-fragmentation process exhibits a percolation transition with a finite-ring phase in which all rings have microscopic length and a giant-ring phase where macroscopic rings account for a finite fraction of the entire mass. Interestingly, while the total mass of the giant rings is a deterministic quantity, their total number and their sizes are stochastic quantities. The size distribution of the macroscopic rings is universal, although the span of this distribution increases with time. Moreover, the average number of giant rings scales logarithmically with system size. We introduce a card-shuffling algorithm for efficient simulation of the ring formation process, and present numerical verification of the theoretical predictions.Comment: 6 pages, 7 figure