The critical behaviour of two-dimensional self-avoiding random walks

1982 ◽  
Vol 48 (3) ◽  
pp. 255-260 ◽  
Author(s):  
P. Grassberger
Keyword(s):  
Random Walks ◽  
Critical Behaviour ◽  
Two Dimensional

scholarly journals Universality class of the motility-induced critical point in large scale off-lattice simulations of active particles.

Soft Matter ◽  
2021 ◽  
Author(s):  
Claudio Maggi ◽  
Matteo Paoluzzi ◽  
Andrea Crisanti ◽  
Emanuela Zaccarelli ◽  
Nicoletta Gnan
Keyword(s):  
Phase Separation ◽  
Critical Point ◽  
Computer Simulations ◽  
Large Scale ◽  
Universality Class ◽  
Critical Behaviour ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Active Particles ◽  
Two Dimensional Model

We perform large-scale computer simulations of an off-lattice two-dimensional model of active particles undergoing a motility-induced phase separation (MIPS) to investigate the systems critical behaviour close to the critical point...

scholarly journals Limit theorems for isotropic random walks on triangle buildings

2002 ◽  
Vol 73 (3) ◽  
pp. 301-334 ◽  
Author(s):  
Marc Lindlbauer ◽  
Michael Voit
Keyword(s):  
Limit Theorem ◽  
Random Walks ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Two Dimensional ◽  
Littlewood Polynomials ◽  
Large Numbers ◽  
Moment Functions ◽  
Vertex Sets ◽  
Isotropic Random

AbstractThe spherical functions of triangle buildings can be described in terms of certain two-dimensional orthogonal polynomials on Steiner's hypocycloid which are closely related to Hall-Littlewood polynomials. They lead to a one-parameter family of two-dimensional polynimial hypergroups. In this paper we investigate isotropic random walks on the vertex sets of triangle buildings in terms of their projections to these hypergroups. We present strong laws of large numbers, a central limit theorem, and a local limit theorem; all these results are well-known for homogeneous trees. Proofs are based on moment functions on hypergroups and on explicit expansions of the hypergroup characters in terms of certain two-dimensional Tchebychev polynimials.

On the critical behaviour of the two dimensional spin XY model

1982 ◽  
Vol 60 (3) ◽  
pp. 368-372 ◽  
Author(s):  
Jos Rogiers
Keyword(s):  
Critical Point ◽  
Triangular Lattice ◽  
Second Order ◽  
Critical Behaviour ◽  
Transverse Magnetization ◽  
Xy Model ◽  
Two Dimensional ◽  
Exponential Behaviour ◽  
Transformation Methods

Transformation methods are used to analyse the series for the second order fluctuation of the transverse magnetization for the triangular and square lattices. For the triangular lattice some evidence is found for an exponential behaviour of this quantity near the critical point with a tentative estimate for the exponent [Formula: see text].

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