We study two families of probability measures on integer partitions, which are Schur measures with parameters tuned in such a way that the edge fluctuations are characterized by a critical exponent different from the generic $1/3$. We find that the first part asymptotically follows a higher-order analogue of the Tracy-Widom GUE distribution, previously encountered by Le Doussal, Majumdar and Schehr in quantum statistical physics. We also compute limit shapes, and discuss an exact mapping between one of our families and the multicritical unitary matrix models introduced by Periwal and Shevitz.
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