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We study the universal scaling limit of random partitions obeying the Schur
measure. Extending our previous analysis [arXiv:2012.06424], we obtain the
higher-order Pearcey kernel describing the multi-critical behavior in the cusp
scaling limit. We explore the gap probability associated with the higher
Pearcey kernel, and derive the coupled nonlinear differential equation and the
asymptotic behavior in the large gap limit.
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