Multi-multifractality and dynamic scaling in stochastic porous lattice

Abstract

In this article, we extend the idea of stochastic dyadic Cantor set to weighted planar stochastic lattice that leads to a stochastic porous lattice. The process starts with an initiator which we choose to be a square of unit area for convenience. We then define a generator that divides the initiator or one of the blocks, picked preferentially with respect to their areas, to divide it either horizontally or vertically into two rectangles of which one of them is removed with probability q = 1 − p. We find that the remaining number of blocks and their mass varies with time as t p and t −q , respectively. Analytical solution shows that the dynamics of this process is governed by infinitely many hidden conserved quantities each of which is a multifractal measure with porous structure as it contains missing blocks of various different sizes. The support where these measures are distributed is fractal with fractal dimension 2p provided 0 < p < 1. We find that if the remaining blocks are characterized by their respective area, then the corresponding block size distribution function obeys dynamic scaling.

Recent development on fragmentation, aggregation and percolation

“…Interestingly, if we replace the center of each cell of the WPSL by a node and common border between cells by a link between the corresponding node then it emerges as a scale-free network. More recently, we have solved a class of models where by dividing the plane vertically or horizontally with equal probability the resulting network is not only scale-free with smaller exponent of the power-law degree distribution but also small-world [56,57]. It is small-world because we find that the mean geodesic path length increases logarithmically with system size and the total mean clustering coefficient is high and independent of system size.…”

Frontiers of fractals for complex systems: recent advances and future challenges

“…For the relative multifractal formalism of non-regular Moran measures, some sufficient conditions are developed [ 22 ]. Further, the idea of a stochastic dyadic Cantor set is extended to a weighted planar stochastic lattice, which leads to a stochastic porous lattice [ 23 ].…”