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- W4298049116 abstract "Random multiplicative processes $w_t =lambda_1 lambda_2 ... lambda_t$ (with < lambda_j > 0 ) lead, in the presence of a boundary constraint, to a distribution $P(w_t)$ in the form of a power law $w_t^{-(1+mu)}$. We provide a simple and physically intuitive derivation of this result based on a random walk analogy and show the following: 1) the result applies to the asymptotic ($t to infty$) distribution of $w_t$ and should be distinguished from the central limit theorem which is a statement on the asymptotic distribution of the reduced variable ${1 over sqrt{t}}(log w_t -< log w_t >)$; 2) the necessary and sufficient conditions for $P(w_t)$ to be a power law are that <log lambda_j > < 0 (corresponding to a drift $w_t to 0$) and that $w_t$ not be allowed to become too small. We discuss several models, previously unrelated, showing the common underlying mechanism for the generation of power laws by multiplicative processes: the variable $log w_t$ undergoes a random walk biased to the left but is bounded by a repulsive ''force''. We give an approximate treatment, which becomes exact for narrow or log-normal distributions of $lambda$, in terms of the Fokker-Planck equation. 3) For all these models, the exponent $mu$ is shown exactly to be the solution of $langle lambda^{mu} rangle = 1$ and is therefore non-universal and depends on the distribution of $lambda$." @default.
- W4298049116 created "2022-10-01" @default.
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- W4298049116 date "1996-09-06" @default.
- W4298049116 modified "2023-09-26" @default.
- W4298049116 title "Convergent multiplicative processes repelled from zero: power laws and truncated power laws" @default.
- W4298049116 doi "https://doi.org/10.48550/arxiv.cond-mat/9609074" @default.
- W4298049116 hasPublicationYear "1996" @default.
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