Positivity, canonical mixing decompositions, and rigorous Whittaker scaling limits in the classification of nearest-neighbor Markov dynamics on Macdonald processes

Borodin and Petrov (Adv. Math. 2016) classified the continuous-time nearest neighbor 'dynamics' on interlacing arrays that act nicely on ascending Macdonald processes: every such 'dynamics' is a mixing of $N!+2(N-1)!+1$ fundamental ones. The classification is algebraic and leaves three structural gaps that the authors themselves flagged. First, positivity is understood only in specializations: all fundamental 'dynamics' are honest Markov processes at $q=t$ (Schur) and exactly three at $t=0$ ($q$-Whittaker), while only the push-block dynamics is known to be positive for generic Macdonald parameters $(q,t)$. Second, the mixing decomposition of an arbitrary nearest neighbor 'dynamics' is far from unique, and no canonical or minimal (linear-in-$N$) parametrization of the class is known. Third, the $q\nearrow1$ scaling limits of the $q$-Whittaker multivariate dynamics to the diffusions on $\mathbb{R}^{N(N+1)/2}$ that couple Whittaker processes, in which the free-energy stochastic differential equations (SDEs) of the semi-discrete directed polymer arise as the subsystems governing the rightmost particle of each level — rest on an 'empiric argument' rather than a convergence proof, though the push-block ('symmetric') case was considered earlier (ref. 2, Thm. 4.1.27). This problem asks to close these three gaps, which jointly complete the probabilistic theory of the classified family.

Background

Macdonald symmetric functions $P_\lambda(\cdot;q,t)$ are a two-parameter family of symmetric polynomials indexed by partitions $\lambda$, with $0\le q,t<1$, that deform the Schur functions: at $q=t$ they reduce to Schur functions, at $t=0$ to the $q$-Whittaker functions, and at $q=0$ to the Hall–Littlewood functions. A Macdonald-nonnegative specialization is an algebra homomorphism from the algebra of symmetric functions to $\mathbb{C}$ determined by nonnegative parameters; the ascending Macdonald process $\mathcal{M}_{\mathrm{asc}}(a_1,\dots,a_N;\rho)$, built from the skew Cauchy identity for Macdonald functions with positive variables $a_1,\dots,a_N$, is the induced probability measure on interlacing arrays $\boldsymbol\lambda=\{\lambda^{(k)}_j\colon 1\le j\le k\le N\}$ — triangular arrays of integers, also called Gelfand–Tsetlin patterns, whose adjacent levels satisfy the constraints $\lambda^{(k)}_{j+1}\le\lambda^{(k-1)}_j\le\lambda^{(k)}_j$. In the degeneration $q=t$ these measures become versions of the Schur process of Okounkov and Reshetikhin.

Borodin and Petrov asked which continuous-time Markov processes on such arrays 'act nicely' on ascending Macdonald processes. A nearest neighbor dynamics moves particles one unit to the right, a move at one level may immediately trigger moves at the level above (a push or pull), and the process must be Markov and map $\mathcal{M}_{\mathrm{asc}}(a_1,\dots,a_N;\rho)$ to $\mathcal{M}_{\mathrm{asc}}(a_1,\dots,a_N;\rho,\rho_\tau)$ after time $\tau$, where $\rho_\tau$ is the Plancherel specialization (the specialization with a single nonnegative $\gamma$-parameter $\tau$): time evolution of the dynamics corresponds to appending that specialization. Dropping the requirement that jump rates and triggered-move probabilities be nonnegative, one speaks of formal 'dynamics' (the quotation marks flag the missing positivity), and the classification problem becomes algebraic: the rates $w_m$ of independent jumps and the probabilities $r_j$ and $c_j$ of long-range pushes and of move propagation on each slice of the array satisfy an explicit linear system whose coefficients $S_j$, $T_i$ are rational functions of $q$, $t$, and the gaps between particles.

The general solution of this system is spanned by finitely many fundamental solutions, unified by the authors into $N!+2(N-1)!+1$ fundamental 'dynamics': one push-block dynamics (a moving particle pushes its upper right neighbor only when the destination is blocked, so all interaction is short-range; it first appeared in the work of Borodin and Corwin on Macdonald processes), $N!$ RSK-type 'dynamics' (at each level only one designated particle jumps independently, all other moves are triggered), and $(N-1)!$ right-pushing and $(N-1)!$ left-pulling 'dynamics' (exactly one long-range push, respectively pull, per level). The main theorem of the paper is that every nearest neighbor 'dynamics' is a mixing — a state-dependent linear combination, with coefficients $\theta^{(k)}(\nu^{(k-1)},\lambda^{(k)})$ attached to consecutive slices of the array, of the fundamental 'dynamics'.

Positivity is only partially understood. The push-block dynamics has nonnegative rates and triggered-move probabilities for all $(q,t)$. In the Schur case ($q=t$) all $N!+2(N-1)!+1$ fundamental 'dynamics' are honest (nonnegative) Markov processes, and the $N!$ RSK-type ones become deterministic insertion algorithms — relatives of the classical Robinson–Schensted correspondence, the bijection between words and pairs of Young tableaux built from row or column insertion — of which only the row and column versions were previously known, leaving $N!-2$ new insertion algorithms. In the $q$-Whittaker case ($t=0$) exactly three fundamental 'dynamics' are honest Markov processes: the push-block, the RSK-type dynamics with designation vector $\boldsymbol h=(1,\dots,1)$ (whose rightmost particles form the $q$-PushTASEP, an interacting particle system in which each particle jumps right at its own rate and pushes its right neighbor with probability $q$ raised to the gap between them, pushes propagating further), and the right-pushing dynamics with $\boldsymbol h=(1,\dots,1)$. For generic $(q,t)$, outside these specializations, the paper determines nothing; and its remark that for $t\ne0$ the coefficient $S_k$ depends on all particles of both adjacent levels — ruling out any $q$-PushTASEP-type Markovian evolution of the edge particles — suggests strong rigidity.

The $q\nearrow1$ regime links the classification to Brownian polymer models. Under the scaling $q=e^{-\varepsilon}$, $\tau=\varepsilon^{-2}T$, $a_k=e^{-\varepsilon\alpha_k}$, with coordinates recentred as $\lambda^{(k)}_j=T\varepsilon^{-2}-(k+1-2j)\varepsilon^{-1}\log\varepsilon+x^{(k)}_j\varepsilon^{-1}$, the $q$-Whittaker processes converge as measures to Whittaker processes: probability measures on $\mathbb{R}^{N(N+1)/2}$ expressed through $SL_{N+1}$ Whittaker functions and tied to the partition function of the semi-discrete directed polymer (up-right paths in $\mathbb{R}\times\mathbb{Z}$ through independent Brownian motions). The free energies of that polymer satisfy a system of stochastic differential equations (SDEs) identified by O'Connell, and Borodin and Petrov formally associate to each $q$-Whittaker multivariate 'dynamics' a system of SDEs for a diffusion on $\mathbb{R}^{N(N+1)/2}$ that couples Whittaker processes in the same 'nice' way; under the identification $\beta_k=-\alpha_k$, the polymer free-energy SDEs arise as the subsystems governing the rightmost particle of each level in the push-block system and in the right-pushing and RSK-type systems with designation vector $(1,\dots,1)$. The derivation, however, is an 'empiric argument' — Poisson increments becoming Brownian and a first-order expansion of the push probabilities — not a convergence theorem; the push-block ('symmetric') case aside, no convergence proof is known for the fundamental 'dynamics' and their mixings.

The unresolved core of this picture is threefold: which formal 'dynamics' are honest Markov processes at generic $(q,t)$; how to remove the non-uniqueness of the mixing decomposition; and whether the formal $q\nearrow1$ convergence of the $q$-Whittaker dynamics to the Whittaker diffusions can be made rigorous.

Problem Statement

Close the three structural gaps that Borodin and Petrov identified in their classification of nearest neighbor dynamics on ascending Macdonald processes (Adv. Math. 300 (2016) 71–155, arXiv:1305.5501). The three components concern one and the same classified family of dynamics and jointly complete its probabilistic theory — which formal 'dynamics' are genuine Markov processes, how the class is parametrized, and what its continuum limits are. All three components are required: resolving any single component alone does not close the problem.

(1) Positivity at generic Macdonald parameters. Determine, for arbitrary $(q,t)\in[0,1)^2$ (in particular outside the already-treated specializations $q=t$ and $t=0$), which of the $N!+2(N-1)!+1$ fundamental 'dynamics' — and, more generally, which solutions of the linear system for the parameters $(w_m,r_j,c_j)$ and their mixings — have nonnegative jump rates and triggered-move probabilities bounded in $[0,1]$ (that is, $w_m\ge0$ and $0\le r_j\le c_j\le1$ throughout), and hence define honest Markov processes on interlacing arrays. The answer must decide, among other things, whether the push-block dynamics is the only fundamental 'dynamics' that is positive for generic $(q,t)$.

(2) Canonical parametrization of the class. The decomposition of an arbitrary nearest neighbor 'dynamics' as a mixing of fundamental 'dynamics' is far from unique. Produce a canonical parametrization of the class: an explicit reduced generating family of fundamental 'dynamics' of size linear in $N$, together with a uniqueness theorem for the mixing decomposition that states the notion of identity under which the decomposition is unique and covers the degenerations arising when particles of adjacent levels become close.

(3) Rigorous Whittaker scaling limits. Prove, under the scaling $q=e^{-\varepsilon}$, $\tau=\varepsilon^{-2}T$, $a_k=e^{-\varepsilon\alpha_k}$ with the recentred coordinates used in the source, that the $q$-Whittaker multivariate dynamics converge at the level of path measures to the diffusions on $\mathbb{R}^{N(N+1)/2}$ that act on Whittaker processes — the systems of SDEs formally derived by the authors, including the push-block ('symmetric') system, in which the semi-discrete directed polymer free-energy SDEs arise as the subsystems governing the rightmost particle of each level. This covers the fundamental 'dynamics' that are honest Markov processes at $t=0$ and their positive mixings and, as anticipated in the source (its Remark 8.8), also the remaining formal 'dynamics' after $q$-Whittaker specialization, even where the pre-limit 'dynamics' admits negative jump rates or triggered-move probabilities, replacing the source's 'empiric argument' by a rigorous convergence theorem.

Scope note. The source's open discussion also (i) deferred the detailed probabilistic analysis of the $q$-PushTASEP to later work — the program pursued in refs. 3–6 — and (ii) proposed an open-ended combinatorial study of the $N!-2$ new deterministic Schur-case insertion algorithms. Neither is a deliverable of this problem.

References

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