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The Macdonald (q,t)(q,t)-permanent II: Probability laws and their regions of validity

A mix of human-written and AI-generated textHuman understanding: some partsmath.PR — Probabilitymath.CO — Combinatoricsmath.FA — Functional Analysis

Contributed by Suvrit Sra ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

The Macdonald (q,t)(q,t)-permanent replaces the constant cycle weight of the Vere-Jones α\alpha-permanent by (1−tk)/(1−qk)(1-t^k)/(1-q^k) for a cycle of length kk. We determine which of the probability laws built on the α\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∈{0,q,q2,…}t\in\{0,q,q^2,\ldots\}, and in dimension two exactly when t≤qt\le q. We conjecture that in each fixed dimension the matrix law exists exactly on the region that we show to be necessary, a qq-analogue of Gindikin's theorem.

Provenance statement

The results in this paper were developed through extensive use of an AI assistant (OpenAI's GPT Pro). None of the claims were obtained zero-shot: they emerged through sustained back-and-forth discussion, including failed attempts, counterexamples, audits, and repairs. The expansion of the author's notes into the present manuscript, together with its finite symbolic verification program, also used OpenAI's models, and Anthropic's Claude (Fable 5.1 and Opus 5.5) prepared this version of the manuscript. Assistant checks and reruns are not external peer review or formal verification. The main text has not yet been vetted line by line by the author.

Tools used

OpenAI
GPT Pro
Anthropic
Claude Opus 5.5 and Fable 5.1

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Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 31, 2026