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Parity-dependent localization in $N$ strongly coupled chains

Anderson localization of wave-functions at zero energy in quasi-1D systems of $N$ disordered chains with inter-chain coupling $t$ is examined. Localization becomes weaker than for the 1D disordered chain ($t=0$) when $t$ is smaller than the longitudinal hopping $t'=1$, and localization becomes usually much stronger when $t\gg t'$. This is not so for all $N$. We find "immunity" to strong localization for open (periodic) lateral boundary conditions when $N$ is odd (a multiple of four), with localization that is weaker than for $t=0$ and rather insensitive to $t$ when $t \gg t'$. The peculiar $N$-dependence and a critical scaling with $N$ is explained by a perturbative treatment in $t'/t$, and the correspondence to a weakly disordered effective chain is shown. Our results could be relevant for experimental studies of localization in photonic waveguide arrays.