Discrete Whittaker Processes

• Discrete Whittaker processes are Markov processes on non-negative integer arrays with interlacing constraints, serving as the combinatorial counterpart to SL(r+1) Whittaker functions and the quantum Toda lattice.
• They evolve via subtractive local Poisson dynamics and recursive Whittaker functions, enabling exact integrability through Doob transforms and intertwinings with lower-dimensional projections.
• Characteristic for their unique entrance laws from infinity and hierarchical Gibbs–Markov structures, these processes connect deeply with growth models, directed polymers, and quantum integrable systems.

A discrete Whittaker process is a Markov process on non-negative integer arrays (typically triangular or skew-shaped) subject to specific interlacing constraints that arise as a discrete, combinatorial counterpart to $SL(r+1, \mathbb{R})$ Whittaker functions and the quantum Toda lattice. In its canonical form, the process is characterized by subtractive, local Poisson-type dynamics on array entries, exact integrability via recursive Whittaker functions, rich Markov projections (notably to boundary data), and exact correspondences with growth models, directed polymers, quantum integrable systems, and special functions. The discrete Whittaker process exhibits remarkable algebraic and probabilistic structure, including intertwining relations, Doob transforms, entrance laws from infinity, and connection to Brownian functionals and Bose–gas-type particle models (O'Connell, 2022).

1. State Space, Array Dynamics, and Interlacing Conditions

The state space of the discrete Whittaker process consists of non-negative integer arrays $\pi = (\pi_{i,j})$ indexed by $1 \leq i, j$ with $i + j \leq r+1$, with $r \geq 1$ being the rank parameter. Each array must satisfy local constraints,

$$\pi_{i,j} \geq \max\Big\{ \pi_{i,j-1}, \pi_{i-1,j} - \beta_{ij} \Big\},$$

where $\beta_{ij}$ is a linear function of the index sequence $\alpha = (\alpha_1, \dots, \alpha_r)$. The prototypical shape is the staircase Young diagram $\delta_{r+1} = (r, \dots, 1)$, for which arrays are precisely reverse plane partitions when $\alpha \equiv 0$.

Boundary conditions are encoded by top-row values: for $n \in \mathbb{N}^r$, the set $\Pi^r_n$ denotes arrays with $\pi_{i, r-i+1} = n_i$ for $i=1, \dots, r$. More generally, the framework extends from the staircase to arbitrary Young diagrams $\lambda$ and subdiagrams $\mu \subset \lambda^\circ$ subject to buffer constraints, encompassing both strictly triangular and skew shapes (O'Connell, 2022).

2. Markov Generator, Fundamental Whittaker Coefficients, and Intertwining

The generator $G^r$ of array-valued dynamics is a discrete non-symmetric Toda-type operator. At site $(i, j)$, the "down-jump" rate is given by

$$b_{ij}(\pi) = (\pi_{ij} - \pi_{i,j-1})\cdot(\pi_{ij} - \pi_{i-1,j} + \beta_{ij}),$$

and the array evolves as a pure death process with generator

$$G^r = \sum_{1 \leq i + j \leq r+1} b_{ij}(\pi) D_{\pi_{ij}}.$$

Here, $D_{\pi_{ij}} f(\pi) = f(\pi - e_{ij}) - f(\pi)$, the backward-difference at $(i, j)$.

The process is constructed to ensure integrability via special coefficients $a_r(n)$, recursively defined: $a_1(n) = \frac{1}{n! (n+\alpha_1)!},$ and for $r \geq 2$,

$$a_r(n) = \sum_{0 \leq k_i \leq n_i} q_r(n, k)a_{r-1}(k),$$

$$q_r(n, k) = \prod_{i=1}^r \frac{1}{(n_i - k_i)! (n_i - k_{i-1} + \alpha_{i r})!},$$

with $k_0 = k_r := 0$.

The coefficients $a_r(n)$ solve the difference quantum Toda eigenrelation

$h^r a_r(n) = 0,$

$$h^r = \sum_{i=1}^r L_{n_i} - \sum_{i=1}^r n_i^2 + \sum_{i=1}^{r-1} n_i n_{i+1} - \sum_{i=1}^r \alpha_i n_i,$$

where $L_k f(k) = f(k-1)$.

By performing a Doob $h$-transform using $a_r(n)$, one obtains a boundary Markov generator

$$L^r = a_r(n)^{-1} \circ h^r \circ a_r(n) = \sum_{i=1}^r [a_r(n-e_i) / a_r(n)] D_{n_i},$$

with strong intertwining between the array and boundary projections via a Gibbs-type kernel $K^r_n(\pi) = w_r(\pi) / a_r(n)$ in terms of inverse-factorial weights $w_r$ (O'Connell, 2022).

3. Entrance Law from Infinity and Uniqueness

A key structural result is the existence and uniqueness of the "entrance law from $+\infty$" for the discrete Whittaker process. By allowing $\pi_{ij} = +\infty$ and considering monotone couplings, the dynamics converge, in the large-boundary limit, to a unique process invariant under time evolution. This translates in the boundary to the $L^r$-Doob-transformed process started from $n(0) \equiv +\infty$, and the conditional law at time $t > 0$ is again given by the Whittaker–Gibbs measure $K^r_{n(t)}$ at the evolving boundary (O'Connell, 2022).

4. Connections to Brownian Functionals, Polymer Models, and Related Integrable Systems

Discrete Whittaker processes are tightly linked to quantum integrable systems, combinatorial growth processes, and stochastic models:

• Quantum Toda duality: The generator $h^r$ is the discrete spectral dual of the quantum Toda Hamiltonian. This duality gives rise to explicit representations of transition probabilities via imaginary Brownian functionals. Specifically, for $p_t(n, m)$ the transition kernel, one has

$$p_t(n, m) = \frac{a_r(m)}{a_r(n)} \frac{1}{(n - m)!} \mathbb{E}\big[Y(t)^{-n} Z(t)^{n-m}\big],$$

with $Y_k(t) = \exp\{ i (B_k - B_{k+1})(t) \}$ for independent real Brownian motions $B_k$, and $Z_k(t) = \int_0^t Y_k(s) ds$ (O'Connell, 2022).

• Imaginary-disordered semi-discrete polymer: Moment functionals of polymer partition sums involving $$U^r(t) = \int_{0 < s_1 < \cdots < s_r < t} Y_1(s_1) \cdots Y_r(s_r) ds_1 \cdots ds_r$$ are naturally dual to the Markov chain on ordered boundary coordinates.
• Corner growth and discrete $\delta$-Bose gas: Embedded within a larger lattice, the process in the vertical coordinate yields a continuous-time box addition growth process, while the evolution of a fixed row is equivalent (after Doob transform) to a discrete repulsive $\delta$-Bose gas.

These connections motivate generalizations to other root systems (e.g., $B_r$, $BC_2$, $G_2$), as well as extensions to more general array shapes with prescribed boundary or buffer conditions (O'Connell, 2022).

5. Markov Projections, Extensions, and Hierarchical Structure

The multilevel, multishape nature of the discrete Whittaker process is encoded by intertwinings between array, boundary, and lower-dimensional projections. The central algebraic identity,

$h^r \circ q_r = q_r \circ h^{r-1},$

establishes hierarchical Gibbs–Markov compatibility and allows for Markov projections to subarrays and skew shapes, as well as natural extensions to other root systems with adjusted combinatorics and transition structures (e.g., shifted staircases for type $B_r$) (O'Connell, 2022).

The process is also robust under transitions to generalized shapes $(\lambda, \mu)$, with explicit combinatorial constructions for the projected generators and their Doob transforms, as well as for initial (entrance) laws.

6. Summary of Structural Results and Key Properties

• The discrete Whittaker process on arrays is governed by a local subtractive dynamic induced by the generator $G^r$.
• The induced boundary evolution is Markovian, with generator $L^r$ and conditional distribution given by the Whittaker–Gibbs measure.
• The key intertwining structure $h^r \circ q_r = q_r \circ h^{r-1}$ implies a consistent multilevel Markov–Gibbs architecture.
• The Whittaker coefficients $a_r(n)$ are both solutions to the discrete Toda difference equation and organized as moments of exponential functionals of Brownian motion, as well as acting as normalization constants for entrance laws.
• Uniqueness of the entrance law from infinity is established via monotone coupling and large deviation analysis for the Gibbs–Whittaker measure $K^r_n$ as $n \to \infty$.
• Specializations to continuous-time geometric RSK with imaginary Brownian input and semi-discrete polymers with imaginary disorder appear as continuum limits of the process (O'Connell, 2022).

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References

• "Discrete Whittaker processes" (O'Connell, 2022)
• "Absorption times for discrete Whittaker processes and non-intersecting Brownian bridges" (O'Connell, 11 Jan 2026)
• "$q$-Deformed Discrete Whittaker Processes" (Ninness, 2 Sep 2025)
• "Matrix Whittaker processes" (Arista et al., 2022)
• "Integral formulas for two-layer Schur and Whittaker processes" (Barraquand, 2024)