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We deal with countable alphabet locally compact random subshifts of finitetype (the latter merely meaning that the symbol space is generated by anincidence matrix) under the absence of Big Images Property and under theabsence of uniform positivity of the transfer operator. We first establish theexistence of random conformal measures along with good bounds for the iteratesof the Perron-Frobenius operator. Then, using the technique of positive conesand proving a version of Bowen's type contraction (see \cite{Bow75}), we alsoestablish a fairly complete thermodynamical formalism. This means that we provethe existence and uniqueness of fiberwise invariant measures (giving rise to aglobal invariant measure) equivalent to the fiberwise conformal measures.Furthermore, we establish the existence of a spectral gap for the transferoperators, which in the random context precisely means the exponential rate ofconvergence of the normalized iterated transfer operator. This latter propertyin a relatively straightforward way entails the exponential decay ofcorrelations and the Central Limit Theorem.

Countable Alphabet Random Subhifts of finite type with weakly positive transfer operator