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Percolation on fractional Brownian Surfaces

Percolation theory is a classical example of simple system but complex physics which has been a long interest to mathematicians and physicists. In the last half a century, a great amount of work has been carried out to understand the critical behavior near pc. The behavior near the critical point is explained by scaling exponents and in different dimension we have been able to compute them (numerically with success and analytically with bit less confidence). There are approches developed using renormalization and field theory also and they are able to provide a reasonable explaination to the observation made. We study this this problem on a fractional Brownian surface with different hurst exponent (H). We occupy the lattice according to the h(x,y) which is height field and study percolation on this. Indeed its a discontinuous phase transition.

fBm Surfaces
Fig: The above figure shows the height field of a fractional Brownian surfaces for H=0.9.

In particular, we focus on the limit H → 0. We use a universal gap-scaling approach to analyse the transition and its characteristics, following ideas developed in Universal Gap Scaling in Percolation(Fan, Meng, Liu & Saberi, Nat. Phys. 16, 455–461, 2020).

References

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