Agglomerative percolation on the Bethe lattice and the triangular cactus

DOI10.1088/1751-8113/46/33/335001zbMATH Open1276.82020arXiv1209.1937OpenAlexW2101259821MaRDI QIDQ2847990

Yup Kim, Huiseung Chae, Soon-Hyung Yook

Publication date: 25 September 2013

Published in: Journal of Physics A: Mathematical and Theoretical (Search for Journal in Brave)

Abstract: We study the agglomerative percolation (AP) models on the Bethe lattice and the triangular cactus to establish the exact mean-field theory for AP. Using the self-consistent simulation method, based on the exact self-consistent equation, we directly measure the order parameter

P_{i n f t y}

and average cluster size

S

. From the measured

P_{i n f t y}

and

S

we obtain the critical exponents

and

g a m m a_{k}

for

k = 2

and 3. Here

and

g a m m a_{k}

are the critical exponents for

P_{i} n f t y

and

S

when the growth of clusters spontaneously breaks the

Z_{k}

symmetry of the

k

-partite graph (Lau, Paczuski, and Grassberger, 2012). The obtained values are

,

g a m m a_{2} = 0.88 (1)

,

, and

g a m m a_{3} = 0.94 (2)

. By comparing these values of exponents with those for ordinary percolation (

and

g a m m a_{i n f t y} = 1

) we also find the inequalities between the exponents, as

and

g a m m a_{i} n f t y > g a m m a_{3} > g a m m a_{2}

. These results quantitatively verify the conjecture that the AP model belongs to a new universality class if

Z_{k}

symmetry is broken spontaneously, and the new universality class depends on

k

[Lau et al., Phys. Rev. E 86, 011118 (2012)].

Full work available at URL: https://arxiv.org/abs/1209.1937

zbMATH Keywords

Bethe lattice

Mathematics Subject Classification ID

Exactly solvable models; Bethe ansatz (82B23) Percolation (82B43)

This page was built for publication: Agglomerative percolation on the Bethe lattice and the triangular cactus

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q2847990)