Large deviation principles for pattern-avoiding permutations, and limit shapes for constrained Mallows permutations

We study Mallows random permutations conditioned to avoid a given pattern $\alpha$ of length 3, for which we find limit shapes in the space of permutons when the bias parameter is of the form $e^{\beta/n}$. Along the way, we provide parametrizations for $\alpha$-avoiding permutons, and establish large deviation principles for uniform $\alpha$-avoiding permutations.