We consider weakly coupled analytic expanding circle maps on the lattice $\integer^d$ (for $d\ge 1$), with small coupling strength $\epsilon$ and coupling between two sites decaying exponentially with the distance. We study the spectrum of the associated (Perron-Frobenius) transfer operators. We give a Fr\'echet space on which the operator associated to the full system has a simple eigenvalue at $1$ (corresponding to the SRB measure $\mu_\epsilon$ previously obtained by Bricmont--Kupiainen [BK1]) and the rest of the spectrum, except maybe for continuous spectrum, is inside a disc of radius smaller than one. For $d=1$ we also construct Banach spaces of densities with respect to $\mu_\epsilon$ on which perturbation theory, applied to the difference of fixed high iterates of the normalised coupled and uncoupled transfer operators, yields localisation of the full spectrum of the coupled operator (i.e., the first spectral gap and beyond). As a side-effect, we show that the whole spectra of the truncated coupled transfer operators (on bounded analytic functions) are $\OO(\epsilon)$-close to the truncated uncoupled spectra, uniformly in the spatial size. Our method uses polymer expansions and also gives the exponential decay of time-correlations for a larger class of observables than those considered in [BK1].
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