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Theorem 1.2. Proof of Theorem 1.1. We will use the following auxiliary results in the proof.. Let SvwT denote the graph obtained from disjoint graphs S, T by adding an edge joining
While Chan and Li’s proof of Horn’s Theorem and our proof of Mirsky’s Theorem have a similar structure, there is a significant difference: The matrices whose existence is guaranteed
A classical result in the theory of continuous-time stationary Gaussian processes gives sufficient conditions for the equivalence of the laws of two centered processes with
This follows from the log-concavity of the multidimensional exponential distribution (which is a particular case of Pr´ekopa’s theorem [P]; cf. also [Bo1] for a general theory
In section 4 we use this coupling to show the uniqueness of the stationary interface, and then finish the proof of theorem 12. Stochastic compactness for the width of
Our proof of Theorem 1.1 is based on estimates for the structure function, i.e., the spatial Fourier transform of the correlation function with respect to h·i , the
The proof of Theorem 3 is based on Burkholder’s technique, which reduces the problem of proving a martingale inequality to finding a certain spe- cial function.. The description of
The main lemma in the proof of Theorem 1.1 involves estimating the resistances. A combination of the above identities will then yield lower bounds for the hitting times. We then
