[Paper Review] Positive temperature dynamics on Gelfand-Tsetlin patterns restricted by wall

Ioanna Nteka|Warwick Research Archive Portal (University of Warwick)|Feb 20, 2018

Random Matrices and Applications35 references3 citations

TL;DR

This thesis introduces positive temperature dynamics on symplectic Gelfand-Tsetlin patterns restricted by a wall, constructing $q$-deformed stochastic processes via $h$-transforms of random walks and Brownian motions using symplectic Schur functions and $\mathfrak{so}_{2n+1}$-Whittaker functions. The key contribution is a positive temperature analogue of Dyson's Brownian motion of type $B/C$, with convergence of $q$-deformed Whittaker functions to classical ones as $q \to 1$. The process is shown to be an $h$-transform with $h$ given by the symplectic Schur function, and a pathwise connection is established between $q$-deformed polymers and Whittaker processes via integral representations.

ABSTRACT

The thesis focuses on processes on symplectic Gelfand-Tsetlin patterns. In chapter 4, a process with dynamics inspired by the Berele correspondence [Ber86] is presented. It is proved that the shape of the pattern is a Doob $h$-transform of independent random walks with $h$ given by the symplectic Schur function. This is followed by an extension to a $q$-weighted version. This randomised version has itself a branching structure and is related to a $q$-deformation of the $so_{2n+1}$-Whittaker functions. In chapter 5, we present a fully randomised process. This process $q$-deforms a process proposed in [WW09]. In chapter 7 we prove the convergence of the $q$-deformation of the $so_{2n+1}$-Whittaker functions to the classical $so_{2n+1}$-Whittaker functions when $q o 1$. Finally, in chapter 8 we turn our interest to the continuous setting and construct a process on patterns which contains a positive temperature analogue of the Dyson's Brownian motion of type $B/C$. The processes obtained are $h$-transforms of Brownian motions killed at a continuous rate that depends on their distance from the boundary of the Weyl chamber of type $B/C$, with $h$ related with the $so_{2n+1}$-Whittaker functions.

Motivation & Objective

• To develop a positive temperature stochastic dynamics on symplectic Gelfand-Tsetlin patterns restricted by a wall, extending zero-temperature models to finite-temperature settings.
• To construct a $q$-deformed version of the dynamics inspired by the Berele correspondence, preserving a branching structure and linking to $q$-deformations of $\mathfrak{so}_{2n+1}$-Whittaker functions.
• To establish a continuous-time analogue of the $q$-deformed process, yielding a positive temperature version of Dyson’s Brownian motion of type $B/C$.
• To prove the convergence of $q$-deformed $\mathfrak{so}_{2n+1}$-Whittaker functions to their classical counterparts as $q \to 1$.
• To unify probabilistic models with algebraic structures via $h$-transforms, using symplectic Schur functions and Whittaker functions as harmonic functions.

Proposed method

• The dynamics are constructed as Doob $h$-transforms of independent random walks, with $h$ given by the symplectic Schur function, ensuring the process remains within the Weyl chamber of type $B/C$.
• A $q$-weighted version of the process is introduced, preserving a branching structure and related to $q$-deformations of $\mathfrak{so}_{2n+1}$-Whittaker functions.
• The continuous limit is derived by constructing a diffusion process on patterns as an $h$-transform of Brownian motion killed at a rate depending on distance from the boundary of the Weyl chamber.
• The $h$-transform uses the $\mathfrak{so}_{2n+1}$-Whittaker function as the harmonic function, ensuring the process stays within the ordered configuration space.
• The connection between $q$-deformed polymers and Whittaker processes is established via integral representations of partition functions, showing $\mathcal{Z}^N(t) \overset{d}{=} \log \int_0^t e^{\mathcal{Y}^N(s)} ds$, where $\mathcal{Y}^N$ is a Whittaker process.
• Conjectures are formulated for pathwise relations between $B/C$-type Dyson processes and $A$-type Whittaker processes, supported by integral identities and time-reversal arguments.

Experimental results

Research questions

• RQ1 How can positive temperature dynamics be defined on Gelfand-Tsetlin patterns restricted by a wall, generalizing zero-temperature models?
• RQ2 What is the role of the symplectic Schur function as an $h$-function in constructing $h$-transforms of random walks on the Weyl chamber of type $B/C$?
• RQ3 How do $q$-deformations of the dynamics relate to $q$-Whittaker functions of type $B$, and what is their branching structure?
• RQ4 Can a continuous-time analogue of the $q$-deformed process be constructed that corresponds to a positive temperature version of Dyson’s Brownian motion of type $B/C$?
• RQ5 What is the limiting behavior of $q$-deformed $\mathfrak{so}_{2n+1}$-Whittaker functions as $q \to 1$, and how does this relate to the classical Whittaker functions?

Key findings

• The $q$-weighted process on Gelfand-Tsetlin patterns is shown to be an $h$-transform of independent random walks with $h$ equal to the symplectic Schur function, ensuring the process remains within the Weyl chamber.
• The $q$-deformed dynamics preserve a branching structure and are linked to $q$-deformations of $\mathfrak{so}_{2n+1}$-Whittaker functions, extending the connection between integrable probability and special functions.
• A continuous-time process on patterns is constructed as an $h$-transform of Brownian motion killed at a rate depending on distance from the Weyl chamber boundary, with $h$ given by the $\mathfrak{so}_{2n+1}$-Whittaker function.
• The $q$-deformed $\mathfrak{so}_{2n+1}$-Whittaker functions are proven to converge to the classical $\mathfrak{so}_{2n+1}$-Whittaker functions as $q \to 1$, establishing a bridge between discrete and continuous models.
• A pathwise relation is established between $q$-deformed polymers and Whittaker processes: $\mathcal{Z}^N(t) \overset{d}{=} \log \int_0^t e^{\mathcal{Y}^N(s)} ds$, where $\mathcal{Y}^N$ is a Whittaker process, generalizing known results in type $A$ to type $B/C$.
• Conjectures are formulated and supported by integral identities, suggesting that the first coordinate of a $B/C$-type Dyson process is distributed as the logarithm of the integral of the first coordinate of an $A$-type Whittaker process, extending the known zero-temperature result to positive temperature.