Critical Exponents in Long-Range Percolation
Theory and estimation
Y.O. Skabelka (TU Delft - Electrical Engineering, Mathematics and Computer Science)
J. Komjáthy – Mentor (TU Delft - Electrical Engineering, Mathematics and Computer Science)
E. Lorist – Graduation committee member (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Y.M.A. Moreno Alonso – Mentor (TU Delft - Electrical Engineering, Mathematics and Computer Science)
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Abstract
In this thesis we study the long-range
percolation model on Zd, where each pair of vertices x, y
∈ Zd form a
connection with probability 1 − exp(−βJ(x, y)), and J(x, y) decays asymptotically
with the form ∥x − y∥−dα. The parameter α > 0 is fixed, while β
can be varied. This model is an extension of the classical Bernoulli bond
percolation model allowing the modeling of connection phenomena where
long-distance connections play a crucial role. In this paper we study the
critical value βc, the percolation probability θ(β) = Pβ (|K0|
= ∞) and we investigate the critical exponent δ explaining the cluster decay at
criticality. We compile known bounds for the critical exponent δ and derive
foundational results for βc. In the numerical part we simulate
long-range percolation on finite boxes, create new estimators for the
percolation probability θ(β) and studying existing ones for the critical
parameter βc. Using this estimate for βc we estimate the
critical parameter δ using linear regression. The estimators for θ(β) and βc
show consistent stable behaviour conforming to theory. The estimates for
δ in contrast are sensitive to the finite size approximation, showcasing the limits
of simulating critical parameters on a finite scale.
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