The Macdonald $(q,t)$-permanent II: Probability laws and their regions of validity
The Macdonald $(q,t)$-permanent replaces the constant cycle weight of the Vere-Jones $\alpha$-permanent by $(1-t^k)/(1-q^k)$ for a cycle of length $k$. We determine which of the probability laws built on the $\alpha$-permanent survive this deformation, and where. The scalar gamma-type law survives on the whole parameter square, but the matrix law, a compound Wishart matrix, is more fragile. We prove that it exists in every dimension exactly when $t\in\{0,q,q^2,\ldots\}$, and in dimension two exactly when $t\le q$. We conjecture that in each fixed dimension the matrix law exists exactly on…
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