Macdonald processes and measures

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macdonald_processes macdonald_measures macdonald_polynomials

Macdonald measures can be found in [{forrester-rains02}].

Let $m, n$ be positive integers, $a \in [0, 1]^m$, $b \in [0, 1]^n$ and $0 \le q, t \le 1$. Let $P_\lambda$ and $Q_\lambda$ be the Macdonald polynomials, we define the Macdonald measure by

$$\mu_{\text{Mac}} (\lambda) = P_\lambda(a) Q_\lambda(b) \prod_{ij} {(a_j b_i)_\infty \over (a_j b_i t)_\infty}.$$

It is a probability measure due to

1. the Cauchy-Littlewood identity (4.13) in macdonald_polynomials.
2. and the fact that $P_\lambda(a), Q_\lambda(b) \ge 0$ if we plug in the explicit formulas of the Macdonald polynomials.

This is our default Macdonald measure, which we also call the usual Macdonald measure since it involves the usual specialisation of the Macdonald polynomials.

By contrast, here is the dual Macdonald measure, involving the dual specialisation of the Macdonald polynomials.

$$\mu_{\text{dMac}}(\lambda) = P_\lambda(a; q, t) P_{\lambda'} (b; t, q) \prod_{ij}{1 \over 1 + a_j b_i}.$$

TODO: Exponential specialisation case.

For now we only care about the usual case $\mu_{\text{Mac}}$.

Let $\phi$ be the quantity defined in the Pieri rule of Macdonald polynomials. For $u \in [0, 1]$, define kernel $A_u$ by

$$A_u(\lambda, \mu) = \phi_{\mu / \lambda} u^{|\mu| - |\lambda|} \prod_i {(a_i u)_\infty \over (t a_i u)_\infty}$$

Claim. For any $u$, the Macdonald polynomial $P_\lambda(b)$ is $A_u$-harmonic.

Proof. Recall by (2.8)(5.5) of macdonald_polynomials,

$$\sum_n Q_{(n)} (a) u^n = \prod_i {(t a_i u)_\infty \over (a_i u)_\infty}$$

and the Pieri rule $(i)$ of macdonald_polynomials

$$P_\lambda(a) Q_{(r)}(a) = \sum_{\mu: \lambda \prec \mu, |\mu| - |\lambda| = r} \phi_{\mu / \lambda} P_\mu(a).$$

Combining these two we have

$$P_\lambda(a) \prod_i {(t a_i u)_\infty \over (a_i u)_\infty} = \sum_{r \ge 0} P_\lambda(a) Q_{(r)}(a) u^r = \sum_{\mu: \lambda \prec \mu} \phi_{\mu / \lambda} P_\mu(a) u^{|\mu| - |\lambda|}$$

and we are done. $\square$

Definition. Let $h(\lambda) = P_\lambda(a)$. The Macdonald process is a Markov process $X$ with transition kernel $R$ that is the [[doob_transform|Doob-$h$ transform]] of kernel $A_{b_n}$ at time $n$:

$$\prob(X_n = \mu | X_{n - 1} = \lambda, X_{0 : n - 2}) = A_{b_n}(\lambda, \mu) {P_\mu(a) \over P_\lambda(a)}.$$

The Macdonald process induces the Macdonald measure:

Claim. Let $X$ be a Macdonald process with empty initial condition: $X_0 \equiv \emptyset$ is an empty partition. Then for any fixed $n$, $X_n$ distributes according to the Macdonald measure.

Proof. Use the branching formula (7.13) of macdonald_polynomials, we have

$$\prob(X_n = \lambda) = \sum_{\emptyset = \lambda^0 \prec \lambda^1 \prec ... \prec \lambda^n} \left(\prod_i \phi_{\lambda^i / \lambda^{i - 1}} b_i^{|\lambda^i| - |\lambda^{i - 1}|} \right) P_\lambda(a) \prod_{i j} {(b_i a_j)_\infty \over (b_i a_j t)_\infty} = P_\lambda(a) Q_\lambda(b)\prod_{i j} {(b_i a_j)_\infty \over (b_i a_j t)_\infty}.$$

$\square$

References

• [forrester-rains02] Interpretations of some parameter dependent generalizations of classical matrix ensembles, Peter J. Forrester, Eric M. Rains, Probability Theory and Related Fields, Vol. 131, No. 1 2002. [ link ]