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We consider the problem of distinguishing between two hypotheses: that a sequence of signals on a large graph consists entirely of noise, or that it contains the faint trail of a random walker buried in the noise. The problem of computing the error exponent of the optimal detector is simple to formulate, but exhibits deep connections to problems known to be difficult, such as computing Lyapunov exponents of products of random matrices and the free energy density of statistical mechanical systems with quenched disorder. By using random Hamiltonian interpolation, a technique previously applied in the rigorous analysis of spin glass systems, we show that a lower bound for the error exponent can be obtained by studying a generalized version of the random energy model, which is exactly soluble by large deviations techniques. Our bound closely matches the empirical results in numerical experiments, and it suggests a secondorder phase transition phenomenon: below a threshold SNR, the error exponent is nearly constant and near zero, indicating poor performance; above the threshold, there is rapid improvement in performance as the SNR increases. The location of the phase transition depends on the entropy rate of the Markov processes. Finally, I will discuss cases where our lower bound is in fact asymptotically tight, in the limit of large state spaces of the underlying Markov processes. Host: Misha Chertkov 