tailieunhanh - Báo cáo toán học: "Ground State Incongruence in 2D Spin Glasses Revisited"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Ground State Incongruence in 2D Spin Glasses Revisited. | Ground State Incongruence in 2D Spin Glasses Revisited Martin Loebl Department of Applied Mathematics and Institute for Theoretical Computer Science ITI Charles University Malostranske n. 25 118 00 Praha 1 Czech Republic loebl@ Submitted Jan 27 2004 Accepted Jun 11 2004 Published Jun 16 2004 MR Subject Classifications 82D30 05C90 05C80 Abstract A construction supporting a conjecture that different ground state pairs exist in the 2-dimensional Edwards-Anderson Ising spin glass is presented. 1 Introduction A fundamental and extensively studied problem on the way towards understanding the effects of disorder and frustration is to determine the multiplicity of infinite volume groundstates in finite-dimensional realistic models. One conjecture in analogy with the meanfield Sherrington-Kirkpatrick model is that finite-dimensional short-ranged systems with frustration have infinitely many groundstate pairs 2 3 . A different conjecture based on droplet-scaling theories predicts that only one groundstate pair exists 4 5 6 . The simplest system used to study these questions is the Edwards-Anderson Ising model 1 in dimension two. Here the hypothesis that only one groundstate pair exists has received support from seminal analytic work of Newman and Stein 8 7 . The purpose of this paper is to present a combinatorial construction supporting the competing Gratefully acknowledges the support of ICM-P01-05 Project LN00A056 and GAUK 158. This work was completed while visiting the DIM U. Chile. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R40 1 hypothesis. In particular very intuitive Conjectures 4 5 are formulated whose validity implies that incongruent finitely incongruent respectively groundstate pairs exist. These conjectures concern only finite sublattices of the square lattice and hence they may be studied by many tools including computer simulations. The Edwards-Anderson Ising model on a graph G V E is defined by coupling constants Jj assigned to each edge i