Diagonal stationary points of the bethe functional

Grzegorz Siudem; Grzegorz Świątek

doi:10.3934/dcds.2017117

Article Contents

2017, Volume 37, Issue 5: 2717-2743. Doi: 10.3934/dcds.2017117

This issue Previous Article Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems Next Article Topological pressure for the completely irregular set of birkhoff averages

Diagonal stationary points of the bethe functional

Grzegorz Siudem^1, and
Grzegorz Świątek^2, ,

1.
Faculty of Physics, Warsaw University of Technology, Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, PL-00-662 Warsaw, Poland
2.
Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, PL-00-662 Warsaw, Poland

* Corresponding author: g.swiatek@mini.pw.edu.pl
* Corresponding author: g.swiatek@mini.pw.edu.pl

Received: February 29, 2016

Revised: November 30, 2016

Published: May 2017

Both authors supported in part by Narodowe Centrum Nauki - grant 2015/17/B/ST1/00091.

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Abstract

We investigate stationary points of the Bethe functional for the Ising model on a $2$-dimensional lattice. Such stationary points are also fixed points of message passing algorithms. In the absence of an external field, by symmetry reasons one expects the fixed points to have constant means at all sites. This is shown not to be the case. There is a critical value of the coupling parameter which is equal to the phase transition parameter on the computation tree, see [13], above which fixed points appear with means that are variable though constant on diagonals of the lattice and hence the term “diagonal stationary points”. A rigorous analytic proof of their existence is presented. Furthermore, computer-obtained examples of diagonal stationary points which are local maxima of the Bethe functional and hence stable equilibria for message passing are shown. The smallest such example was found on the $100× 100$ lattice.

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Mathematics Subject Classification: Primary: 37N40, 90C26; Secondary: 37A60.

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Figure 1. Illustration of the diagonal matrix which can be obtained from means given by Eq. (11).

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Figure 2. Numerical evidence that the means (from Eq. (11), visualized on Fig. 1) in fact define a stationary point of the Bethe functional. The dots on the graph show values of the negative Bethe functional computed for the means given by vector $B_{\eta}$ given by formula (12) with $\eta$ shown on the horizontal axis and various randomly chosen $(X_{\ell})$.

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Figure 3. Values of the negative Bethe functional for the diagonal stationary point $\mathcal{B}_0$ perturbed in the direction of $P$ according to formula (12).

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Figure 5. The stability test algorithm.

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Figure 4. Stable fixed point given by Eq. (13).

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