Usual-spec Markov-Doob integrability for Macdonald processes

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macdonald_polynomials markov_function_theorem doob_transform macdonald_processes macdonald_measures

We write down usual-spec Markov-Doob integrability equation for Macdonald processes.

In this setting, we have

• \(h(\lambda) = P_\lambda\) are the Macdonald \(P\)-polynomials
• \(B(\lambda, \mu) = \psi_{\mu / \lambda} a_k^{|\mu| - |\lambda|}\)
• \(A_u(\lambda, \mu) = \phi_{\mu / \lambda} u^{|\mu| - |\lambda|} \prod_i {(a_i u)_\infty \over (t a_i u)_\infty}\)
• \(w_{nk} \sim qt\text{Geom}(a_k b_n)\), the qt-geometric distribution.

Theorem. The Markov-Doob integrability equation is

\begin{align} \sum_{\lambda^{n - 1, k - 1}} L( (\lambda^{n - 1, k - 1}, \lambda^{n - 1, k}, \lambda^{n, k - 1}), \lambda^{n, k}) {\psi_{\lambda^{n - 1, k} / \lambda^{n - 1, k - 1}} \phi_{\lambda^{n, k - 1} / \lambda^{n - 1, k - 1}} \over \psi_{\lambda^{n, k} / \lambda^{n, k - 1}} \phi_{\lambda^{n, k} / \lambda^{n - 1, k}}} \notag \\ \times b_n^{|\lambda^{n - 1, k}| + |\lambda^{n, k - 1}| - |\lambda^{n - 1, k - 1}| - |\lambda^{n, k}|} {(t a_k b_n)_\infty \over (a_k b_n)_\infty} = 1. \qquad (1) \end{align}

Proof. Plugging \(A = A_{b_n}\), \(B\) and \(L\) into (1) in Markov-Doob integrability on triangular arrays and simplify. \(\square\)

Combining this with the Corollary in Markov-Doob integrability on triangular arrays and the Claim in Macdonald processes and measures we have

Corollary. If (1) is satisfied then

1. \(\lambda^k|_{\lambda^{0, 1 : k} \sim K(\lambda^{0, k}, \cdot)}\) is the Macdonald process, with transition kernel \[ Q(\lambda, \mu) = P_\lambda(a_{1 : k})^{-1} A_{b_n}(\lambda, \mu) P_\mu(a_{1 : k}), \] which is the Doob h-transform of \(A\).
2. \(\prob(\lambda^{n, 1 : k} = \mu^{1 : k} | \lambda^{n, k} = \mu^k, \lambda^{1 : n - 1, k}) = {\psi_{\mu^{1 : k}} a^{\mu^{1 : k}} \over P_{\mu^k}(a_{1 : k})}.\)
3. The marginal law of \(\lambda^{n, k}\) is the Macdonald measure \[ \prob(\lambda^{n, k} = \mu) = P_\mu(a_{1 : k}) Q_\mu(b_{1 : n}) \prod_{i j} {(b_i a_j)_\infty \over (b_i a_j t)_\infty}. \]