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Singularities in Gaussian random fields. (English) Zbl 0764.60051

Summary: This paper studies a class of Gaussian random fields defined on lattices that arise in pattern analysis. Phase transitions are shown to exist at a critical temperature for these Gaussian random fields. These are established by showing discontinuous behavior for certain field random variables as the lattice size increases to infinity. The discontinuities in the statistical behavior of these random variables occur because the growth rates of the eigenvalues of the inverse of the variance-covariance matrix at the critical temperature are different from the growth rates at noncritical temperatures. It is also shown that the limiting specific heat has a phase transition with a power law behavior. The critical temperature occurs at the end point of the available values of temperature. Thus, although the critical behavior is not extreme, caution should be exercised when using such models near critical temperatures.

MSC:

60G60 Random fields
60G15 Gaussian processes
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References:

[1] Grenander, U., and Sethuraman, J. (1985). Limit theorems in metric pattern theory. (Unpublished). · Zbl 0855.62010
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