On Approximate Pattern Matching for a Class of Gibbs Random Fields by Jean-rene Chazottes,

We prove an exponential approximation for the law of approximate occurrence of typical patterns for a class of Gibssian sources on the lattice Z d , d ≥ 2. From this result, we deduce a law of large numbers and a large deviation result for the waiting time of distorted patterns. 1. Introduction. In recent years there has been growing interest in a detailed probabilistic analysis of pattern matching and approximate pattern matching. For example, in information theory, motivation comes from studying performance of idealized Lempel–Ziv coding schemes. In mathematical biology one likes to have accurate estimates for the probability that two (e.g., DNA) sequences agree in a large interval with some error-percentage. There is also considerable interest in the analysis of occurrence of patterns in the multi-dimensional setting, for example, in the context of video-image compression [2], and more generally, lossy data compression [5, 6, 10]. In this paper we study the following problem. Fix a pattern A n in a cubic box of size n. Given a configuration σ of a Gibbs random field, what is the size of the " observation window " in which we do not necessarily see exactly this pattern for the first time, but any pattern obtained by distortion of the fixed pattern A n ? By this, we mean a pattern which contains a fixed fraction ε of spins different from those of A n. We are interested in the behavior of the volume of this observation window, which we call " approximate hitting-time, " when n grows. Our main result (Theorem 2.6) can be phrased as follows. The distribution of the approximate hitting-time, when properly normalized, gets closer and closer to an exponential law. The normalization is the product of a certain parameter n and the probability of the set of distorted patterns [A n ] ε. In fact, we get a precise control of the error term which allows us to derive two corollaries for the " approximate waiting-time " : given a configuration η randomly chosen from an ergodic Gibbs random field, we increase the observation window in a random configuration σ drawn from the given Gibbs random field until we see approximately the pattern η C n. The first corollary implies a law of large numbers allowing to get the rate-distortion function almost surely from this approximate waiting-time. The