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We compute the joint statistics of the momenta $p_i$ of $N$ non-interactingfermions in a trap, near the Fermi edge, with a particular focus on the largestone $p_{\max}$. For a $1d$ harmonic trap, momenta and positions play asymmetric role and hence, the joint statistics of momenta is identical to thatof the positions. In particular, $p_{\max}$, as $x_{\max}$, is distributedaccording to the Tracy-Widom distribution. Here we show that novel "momentumedge statistics" emerge when the curvature of the potential vanishes, i.e. for"flat traps" near their minimum, with $V(x) \sim x^{2n}$ and $n>1$. These arebased on generalisations of the Airy kernel that we obtain explicitly. Thefluctuations of $p_{\max}$ are governed by new universal distributionsdetermined from the $n$-th member of the second Painlev\'e hierarchy ofnon-linear differential equations, with connections to multicritical randommatrix models. Finite temperature extensions and possible experimentalsignatures in cold atoms are discussed.

Multicritical edge statistics for the momenta of fermions in non-harmonic traps