Bibliographic Details
| Title: |
The Planar Ising Model and Total Positivity. |
| Authors: |
Lis, Marcin1 m.lis@statslab.cam.ac.uk |
| Source: |
Journal of Statistical Physics. Jan2017, Vol. 166 Issue 1, p72-89. 18p. |
| Subjects: |
Alternating current circuits, Planar graphs, Ising model, Nonnegative matrices, Boundary value problems |
| Abstract: |
A matrix is called totally positive (resp. totally nonnegative) if all its minors are positive (resp. nonnegative). Consider the Ising model with free boundary conditions and no external field on a planar graph G. Let $$a_1,\dots ,a_k,b_k,\dots ,b_1$$ be vertices placed in a counterclockwise order on the outer face of G. We show that the $$k\times k$$ matrix of the two-point spin correlation functions is totally nonnegative. Moreover, $$\det M > 0$$ if and only if there exist k pairwise vertex-disjoint paths that connect $$a_i$$ with $$b_i$$ . We also compute the scaling limit at criticality of the probability that there are k parallel and disjoint connections between $$a_i$$ and $$b_i$$ in the double random current model. Our results are based on a new distributional relation between double random currents and random alternating flows of Talaska [37]. [ABSTRACT FROM AUTHOR] |
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| Database: |
Engineering Source |
