{"id":"hexagon/2610.00730","pdf":"https://hexagonmath.org/pdf/2610.00730v1","url":"https://hexagonmath.org/abs/2610.00730v1","tags":["math.CO","math.FA","msc:60E07","msc:60B20","msc:15A15","msc:60G55","msc:05E05","Cox process","Macdonald permanent","Schur measure"],"score":0,"title":"The Macdonald $(q,t)$-permanent II: Probability laws and their regions of validity","domain":"math.PR","origin":"ai-generated","source":"hexagon","authors":["Suvrit Sra"],"reviews":0,"subject":"math.PR","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["GPT Pro","Claude Opus 5.5 and Fable 5.1"],"discussed":0,"published":"2026-11-02","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}