[Paper Review] Pfaffian Stochastic Dynamics of Strict Partitions

Leonid Petrov|arXiv (Cornell University)|Nov 15, 2010

Random Matrices and Applications66 references3 citations

TL;DR

This paper introduces a family of continuous-time Markov jump processes on strict partitions (distinct parts) that preserve Borodin's determinantal measures. Using Pfaffian correlation functions and hypergeometric functions, it derives explicit formulas for both static and dynamical correlation kernels, establishing a novel stationary Pfaffian dynamics distinct from prior determinantal models.

ABSTRACT

We study a family of continuous time Markov jump processes on strict partitions (partitions with distinct parts) preserving the distributions introduced by Borodin (1997) in connection with projective representations of the infinite symmetric group. The one-dimensional distributions of the processes (i.e., the Borodin's measures) have determinantal structure. We express the dynamical correlation functions of the processes in terms of certain Pfaffians and give explicit formulas for both the static and dynamical correlation kernels using the Gauss hypergeometric function. Moreover, we are able to express our correlation kernels (both static and dynamical) through those of the z-measures on partitions obtained previously by Borodin and Olshanski in a series of papers. The results about the fixed time case were announced in the author's note arXiv:1002.2714. A part of the present paper contains proofs of those results.

Motivation & Objective

• To construct continuous-time Markov jump processes on strict partitions that preserve Borodin's probability measures.
• To derive explicit formulas for dynamical correlation functions in terms of Pfaffians.
• To express the static and dynamical correlation kernels using the Gauss hypergeometric function.
• To relate the new kernels to those of the z-measures via a rank-one perturbation structure.
• To establish a first example of a stationary Pfaffian stochastic dynamics in the context of random partitions.

Proposed method

• Utilizes a coherency property of Borodin's measures on strict partitions, analogous to that in Borodin-Olshanski dynamics.
• Models the state space as finite point configurations on the positive integers, with dynamics preserving the measure family.
• Expresses dynamical correlation functions as Pfaffians of a kernel function $\boldsymbol{\Phi}_{\alpha,\xi}(s,x;t,y)$ involving the Gauss hypergeometric function.
• Reduces the Pfaffian structure to determinants via a novel reduction technique using an involution and a strictly positive function $f$.
• Derives the static kernel $\mathbf{K}_{\alpha,\xi}(x,y)$ as a rank-one perturbation of a projection operator in $\ell^2(\mathbb{Z})$.
• Applies a transformation using $SL(2,\mathbb{C})^n$-actions to convert the skew-symmetric Pfaffian matrix into a block form amenable to determinant reduction.

Experimental results

Research questions

• RQ1 Can a stationary Markov process be constructed on strict partitions that preserves Borodin's measures and exhibits Pfaffian correlation structure?
• RQ2 How do the dynamical correlation functions of such processes differ from those in determinantal models like the z-measures?
• RQ3 What is the explicit form of the extended kernel $\boldsymbol{\Phi}_{\alpha,\xi}(s,x;t,y)$ governing the space-time correlations?
• RQ4 How is the static Pfaffian kernel related to the known determinantal kernels of the z-measures?
• RQ5 Can the Pfaffian structure be reduced to a determinant via a suitable transformation, and what conditions ensure this?

Key findings

• The dynamical correlation functions of the Markov processes are expressed as Pfaffians of a kernel $\boldsymbol{\Phi}_{\alpha,\xi}(s,x;t,y)$, explicitly given in terms of the Gauss hypergeometric function.
• The static correlation functions reduce to determinantal form, with the kernel $\mathbf{K}_{\alpha,\xi}(x,y)$ being an explicit hypergeometric-type function on $\mathbb{Z}_{>0}$.
• The static kernel $\mathbf{K}_{\alpha,\xi}$ is not a projection operator, distinguishing it from the z-measure kernels.
• The extended kernel $\boldsymbol{\Phi}_{\alpha,\xi}(s,\cdot;s,\cdot)$ on $\ell^2(\mathbb{Z})$ is a rank-one perturbation of an orthogonal projection operator.
• The Pfaffian kernel is reduced to a determinant via a transformation involving an involution and a strictly positive function $f$, with the resulting kernel $K(u,v)$ given by a rational expression involving $F(u,\hat{v})$ and $f(u)$.
• The model provides the first example of a stationary (time-homogeneous) Pfaffian stochastic dynamics in the context of random partitions, contrasting with previous time-inhomogeneous models.