statistical mechanics - Ising model on lattices with (vertical side length) $neq$ (horizontal side length)

Consider the Ising model with nearest neighbours interactions on a rectangular lattice $L\times M$.

If $L=M$ (2-dimensional square lattice), it is known (e.g. by Peierls argument or Onsager explicit solution) that the model exhibits a phase transition when $L=M\to\infty$.

If instead we fix $L=1$ (1-dimensional line) and let $M\to\infty$, the model does not exhibit a phase transition.

My question is: which relation among the side lenghts $L,M$ guarantes the presence/absence of a phase transition? For example what about the case $L=\log M$ ?