Exact Results for the One-Dimensional Self-Organized Critical Forest-Fire Model

We present the analytic solution of the self-organized critical (SOC) forest-fire model in one dimension proving SOC in systems without conservation laws by analytic means. Under the condition that the system is in the steady state and very close to the critical point, we calculate the probability that a string of $n$ neighboring sites is occupied by a given configuration of trees. The critical exponent describing the size distribution of forest clusters is exactly $\tau = 2$ and does not change under certain changes of the model rules. Computer simulations confirm the analytic results.Comment: 12 pages REVTEX, 2 figures upon request, dro/93/

Different hierarchy of avalanches observed in the Bak-Sneppen evolution model

We introduce a new quantity, average fitness, into the Bak-Sneppen evolution model. Through the new quantity, a different hierarchy of avalanches is observed. The gap equation, in terms of the average fitness, is presented to describe the self-organization of the model. It is found that the critical value of the average fitness can be exactly obtained. Based on the simulations, two critical exponents, avalanche distribution and avalanche dimension, of the new avalanches are given.Comment: 5 pages, 3 figure

Conformal field theory correlations in the Abelian sandpile mode

We calculate all multipoint correlation functions of all local bond modifications in the two-dimensional Abelian sandpile model, both at the critical point, and in the model with dissipation. The set of local bond modifications includes, as the most physically interesting case, all weakly allowed cluster variables. The correlation functions show that all local bond modifications have scaling dimension two, and can be written as linear combinations of operators in the central charge -2 logarithmic conformal field theory, in agreement with a form conjectured earlier by Mahieu and Ruelle in Phys. Rev. E 64, 066130 (2001). We find closed form expressions for the coefficients of the operators, and describe methods that allow their rapid calculation. We determine the fields associated with adding or removing bonds, both in the bulk, and along open and closed boundaries; some bond defects have scaling dimension two, while others have scaling dimension four. We also determine the corrections to bulk probabilities for local bond modifications near open and closed boundaries.Comment: 13 pages, 5 figures; referee comments incorporated; Accepted by Phys. Rev.

Spatial-temporal correlations in the process to self-organized criticality

A new type of spatial-temporal correlation in the process approaching to the self-organized criticality is investigated for the two simple models for biological evolution. The change behaviors of the position with minimum barrier are shown to be quantitatively different in the two models. Different results of the correlation are given for the two models. We argue that the correlation can be used, together with the power-law distributions, as criteria for self-organized criticality.Comment: 3 pages in RevTeX, 3 eps figure

Avalanche Merging and Continuous Flow in a Sandpile Model

A dynamical transition separating intermittent and continuous flow is observed in a sandpile model, with scaling functions relating the transport behaviors between both regimes. The width of the active zone diverges with system size in the avalanche regime but becomes very narrow for continuous flow. The change of the mean slope, Delta z, on increasing the driving rate, r, obeys Delta z ~ r^{1/theta}. It has nontrivial scaling behavior in the continuous flow phase with an exponent theta given, paradoxically, only in terms of exponents characterizing the avalanches theta = (1+z-D)/(3-D).Comment: Explanations added; relation to other model

Exact equqations and scaling relations for f-avalanche in the Bak-Sneppen evolution model

Infinite hierarchy of exact equations are derived for the newly-observed f-avalanche in the Bak-Sneppen evolution model. By solving the first order exact equation, we found that the critical exponent which governs the divergence of the average avalanche size, is exactly 1 (for all dimensions), confirmed by the simulations. Solution of the gap equation yields another universal exponent, denoting the the relaxation to the attractor, is exactly 1. We also establish some scaling relations among the critical exponents of the new avalanche.Comment: 5 pages, 1 figur

Noncommutative Vortex Solitons

We consider the noncommutative Abelian-Higgs theory and investigate general static vortex configurations including recently found exact multi-vortex solutions. In particular, we prove that the self-dual BPS solutions cease to exist once the noncommutativity scale exceeds a critical value. We then study the fluctuation spectra about the static configuration and show that the exact non BPS solutions are unstable below the critical value. We have identified the tachyonic degrees as well as massless moduli degrees. We then discuss the physical meaning of the moduli degrees and construct exact time-dependent vortex configurations where each vortex moves independently. We finally give the moduli description of the vortices and show that the matrix nature of moduli coordinates naturally emerges.Comment: 22 pages, 1 figure, typos corrected, a comment on the soliton size is adde

d_c=4 is the upper critical dimension for the Bak-Sneppen model

Numerical results are presented indicating d_c=4 as the upper critical dimension for the Bak-Sneppen evolution model. This finding agrees with previous theoretical arguments, but contradicts a recent Letter [Phys. Rev. Lett. 80, 5746-5749 (1998)] that placed d_c as high as d=8. In particular, we find that avalanches are compact for all dimensions d<=4, and are fractal for d>4. Under those conditions, scaling arguments predict a d_c=4, where hyperscaling relations hold for d<=4. Other properties of avalanches, studied for 1<=d<=6, corroborate this result. To this end, an improved numerical algorithm is presented that is based on the equivalent branching process.Comment: 4 pages, RevTex4, as to appear in Phys. Rev. Lett., related papers available at http://userwww.service.emory.edu/~sboettc

Distribution of repetitions of ancestors in genealogical trees

We calculate the probability distribution of repetitions of ancestors in a genealogical tree for simple neutral models of a closed population with sexual reproduction and non-overlapping generations. Each ancestor at generation g in the past has a weight w which is (up to a normalization) the number of times this ancestor appears in the genealogical tree of an individual at present. The distribution P_g(w) of these weights reaches a stationary shape P_\infty(w) for large g, i.e. for a large number of generations back in the past. For small w, P_\infty(w) is a power law with a non-trivial exponent which can be computed exactly using a standard procedure of the renormalization group approach. Some extensions of the model are discussed and the effect of these variants on the shape of P_\infty(w) are analysed.Comment: 20 pages, 5 figures included, to appear in Physica

Breakdown of self-organized criticality

We introduce two sandpile models which show the same behavior of real sandpiles, that is, an almost self-organized critical behavior for small systems and the dominance of large avalanches as the system size increases. The systems become fully self-organized critical, with the critical exponents of the Bak, Tang and Wiesenfeld model, as the system parameters are changed, showing that these systems can make a bridge between the well known theoretical and numerical results and what is observed in real experiments. We find that a simple mechanism determines the boundary where self-organized can or cannot exist, which is the presence of local chaos.Comment: 3 pages, 4 figure