Multipoint scaling limits, Pfaffian structure, and the Whittaker degeneration chain for half-space Macdonald processes

Half-space Macdonald processes, introduced by Barraquand–Borodin–Corwin (2020), are probability measures on integer partitions built from the Macdonald Littlewood summation identity; their degenerations describe half-space versions of the log-gamma directed polymer, the stochastic six-vertex model, ASEP, and the KPZ equation with a boundary. The source paper proved exact Laplace-transform formulas but left its asymptotic theory formal: the series-of-integrals formulas lack the uniform summability needed for rigorous limit theorems, a Pfaffian reorganization was known only in one special Hall-Littlewood case, the multilevel passage from $q$-Whittaker to Whittaker stochastic-differential dynamics was heuristic, and only one-point asymptotics were available. Later work resolved the one-point goal by an alternative route (Imamura–Mucciconi–Sasamoto proved the Tracy–Widom GSE/GOE/Gaussian Baik–Rains phase diagram via free boundary Schur measures) and built the multipoint theory at zero temperature. The problem posed here combines the surviving core of the source's program — determine whether the general Macdonald-level series admit Pfaffian or determinantal reorganization (in particular a direct identity with the Fredholm–Pfaffian formulas), and close the remaining rigor gaps in the $q\to1$ degeneration chain (multilevel Whittaker dynamics, uniform removal of the Plancherel component) — with the multipoint, process-level convergence of positive-temperature half-space KPZ models to half-space Airy-type line ensembles, the open strengthening explicitly identified by the subsequent literature.

Background

Macdonald symmetric functions $P_\lambda(x;q,t)$ are a two-parameter family of symmetric polynomials indexed by integer partitions $\lambda$, which degenerate, for special values of the parameters, to the Schur functions ($q=t$), the Hall–Littlewood functions ($q=0$), and the $q$-Whittaker functions ($t=0$). Macdonald processes (Borodin–Corwin 2014) are probability measures on chains of partitions built from the Cauchy summation identity for these functions; choosing parameters and specializations appropriately turns expectations of natural observables into exactly solvable computations, which have yielded rigorous fluctuation theorems for interacting particle systems, directed polymers, and the Kardar–Parisi–Zhang (KPZ) stochastic partial differential equation. A specialization is a way of feeding variables and extra parameters into a symmetric function; a Plancherel specialization contributes a single parameter $\gamma$ that later plays the role of an O'Connell–Yor-type polymer component.

The source paper (Barraquand–Borodin–Corwin, Forum of Mathematics Pi 2020) developed the half-space analogue: measures $\mathbb{P}^{q,t}(\lambda)$ on single partitions built from a Macdonald version of the Littlewood summation identity instead of the Cauchy identity, together with half-space Macdonald processes on interlacing chains, and Markovian dynamics preserving them. The half-space geometry carries a boundary: in the $q$-Whittaker degeneration ($t=0$), taking $q\to1$ with suitable scaling produces the half-space log-gamma polymer — the point-to-point partition function $Z^{\mathrm{hs}}(N,N)$ of a directed polymer on the half-quadrant $\{(i,j)\in\mathbb{Z}_{>0}^2: i\le j\}$ with independent inverse-Gamma bulk weights $\Gamma^{-1}(2\theta)$ off the diagonal and $\Gamma^{-1}(\theta+\alpha)$ on it — whose free energy should converge, under KPZ universality, to the Tracy–Widom GSE distribution when the boundary strength $\alpha>0$ is fixed (supercritical regime), to the GOE distribution at the critical point $\alpha=0$, to a Gaussian law when $-\theta<\alpha<0$ (bound/subcritical regime), and to the Baik–Rains crossover distribution when $\alpha$ is scaled like $N^{-1/3}$ near $0$. Here GSE and GOE denote the Tracy–Widom fluctuation laws of the largest eigenvalue of random matrices from the Gaussian symplectic and Gaussian orthogonal ensembles, and the Baik–Rains crossover family interpolates between the GOE and GSE laws. The same phase diagram governs the half-space KPZ equation, that is, the KPZ stochastic PDE $\partial_t h=\tfrac12\Delta h+\tfrac12(\partial_x h)^2+\xi$ on the half-line with a Neumann-type boundary condition $\partial_x h(t,0)=A$.

The analytic core of the source paper consists of two types of exact formulas for Laplace transforms of observables such as the first part $\lambda_1$: (i) formulas in terms of Fredholm–Pfaffian-friendly objects in special degenerations, and (ii) for general $(q,t)$, formulas as a finite but growing series of nested contour integrals of increasing dimension (Theorems 3.12 and 3.20 of the source), whose summands contain gamma-function cross-products that grow super-exponentially in the expansion order. A Fredholm Pfaffian is the Pfaffian analogue of a Fredholm determinant: the expectation of a multiplicative functional of a Pfaffian point process (a determinantal point process whose joint intensities are Pfaffians), and such representations are the standard input for rigorous saddle-point asymptotics. The authors performed a formal steepest-descent analysis of the series-of-integrals, recovering the expected full phase diagram, but could not prove uniform summability of the series as the system size grows — they explicitly noted that this suggests missing structural cancellations hidden in the formulas, and their introduction observes that, if the gamma-function cross-products admitted suitable control, the series could be recast as the Fredholm Pfaffian of an explicit kernel.

One structural cancellation was known: in a special half-space Hall–Littlewood degeneration related to the stochastic six-vertex model in a half-quadrant (a stochastic lattice model of arrows on a square grid, with each of the six allowed vertex configurations carrying a probability weight), the subsequent joint work BBCW18 (Barraquand–Borodin–Corwin–Wheeler, Duke Math. J. 2018) expressed the relevant Laplace transform through the Pfaffian Schur process, a Pfaffian point process introduced by Baik–Rains and used by Baik–Barraquand–Corwin–Suidan for half-space last-passage percolation. Whether analogous Pfaffian or determinantal structure exists at the general Macdonald level, and what mechanism produces it, was left completely open. Two further gaps in the $q\to1$ chain were left open: the convergence of the $q$-Whittaker Markov dynamics to explicit stochastic-differential-equation (SDE) dynamics for the entire triangular (Gelfand–Tsetlin-type) array of interlacing partitions under Plancherel regularization — only the first-coordinate marginal had been justified, via Borodin–Petrov's nearest-neighbor dynamics on Macdonald processes — and the uniform justification of removing the Plancherel component ($\tau\downarrow0$) in all parameter regimes and for multipoint observables.

Finally, the source paper obtained only one-point (formal) distributions: its own asymptotic program targeted the one-point limit laws for $\log Z^{\mathrm{hs}}(N,N)$, which it explicitly flagged as "not completely proved", noting that making them rigorous constitutes a significant challenge. Its introduction mentions multipoint fluctuations — convergence to Airy-type processes and line ensembles — among the remaining open challenges of the Macdonald-process program, there in the full-space context. The multipoint counterpart of one-point Tracy–Widom asymptotics is convergence of rescaled space-time processes to Airy-type processes and line ensembles: the (full-space) Airy line ensemble is the universal infinite family of locally Brownian curves describing edge fluctuations of determinantal point processes, and KPZ-class models are expected to converge to it under the 1:2:3 KPZ scaling (time scaled by $t$, space by $t^{2/3}$, fluctuations by $t^{1/3}$). Subsequent literature proved the one-point phase diagram by an alternative route and constructed half-space Airy line ensembles at zero temperature, but the half-space multipoint question at positive temperature — posed explicitly by Imamura–Mucciconi–Sasamoto (whose correspondence methods "only work for the study of 'single point' functions") and left open by Barraquand–Corwin–Das and Das–Serio (who prove tightness but not convergence) — together with the general Macdonald-level structural question and the two degeneration-chain gaps, remains open; that surviving core is the problem posed here.

Problem Statement

The common objective is to complete the rigorous limit theory of half-space Macdonald processes at the multipoint (process) level and to settle the structural and degeneration-chain questions the source paper left open, given that the one-point phase diagram has since been proved by other methods. Concretely, with the conventions of the source paper (bulk inverse-Gamma weights $\Gamma^{-1}(2\theta)$, boundary weights $\Gamma^{-1}(\theta+\alpha)$ with $\theta>0$ and $\alpha> -\theta$; critical point $\alpha=0$; supercritical $\alpha>0$ fixed; subcritical/bound phase $-\theta<\alpha<0$):

(a) Primary deliverable — process-level convergence at positive temperature: prove that the half-space log-gamma polymer line ensemble of Barraquand–Corwin–Das (the line ensemble whose top curve is the scaled free-energy process of the half-space log-gamma polymer, its lower curves extending the picture away from the diagonal) and the half-space KPZ line ensemble of Das–Serio (the analogous ensemble whose top curve is the scaled Cole–Hopf solution of the half-space KPZ equation) converge — not merely are tight — under the 1:2:3 KPZ scaling, as the system size (respectively time) tends to infinity, in the critical regime (boundary parameter scaled as $\alpha=\mu N^{-1/3}$, respectively $\alpha=\mu t^{-1/3}$ for the half-space KPZ equation) to the critical half-space Airy line ensembles of Dimitrov–Yang, and in the supercritical regime ($\alpha>0$ fixed) to the pinned half-space Airy line ensemble of Dimitrov–Serio–Yang. Multipoint fixed-time convergence statements (joint distributions at several spatial points converging to the Pfaffian point processes with Baik–Rains crossover kernels) may be supplied in addition to, but do not by themselves satisfy, (a).

(b) Structural deliverable — cancellations at the level of the source's formulas: for the general $(q,t)$ half-space Macdonald measure, either prove the uniform summability (tail control of the integrals and of the summation over the expansion order, uniformly as the system size tends to infinity) needed to make the saddle-point analysis of the series-of-integrals Laplace transforms (Theorems 3.12 and 3.20 of the source) rigorous, or establish a Pfaffian or determinantal reorganization of these series, or prove a direct identity between these series and the Fredholm–Pfaffian representations obtained via the free boundary Schur correspondence (the matching, introduced by Imamura–Mucciconi–Sasamoto, between the half-space $q$-Whittaker measure and a Schur measure with a free boundary, whose marginals form Pfaffian point processes) in the corresponding degenerations — the identity that Imamura–Mucciconi–Sasamoto explicitly pose as open relative to the explicit series formulas of the source paper and of O'Connell–Seppäläinen–Zygouras.

(c) Degeneration-chain deliverable — multilevel Whittaker dynamics: prove that the $q$-Whittaker dynamics constructed in the source paper converge, under the $q\to1$ limit with Plancherel regularization, to explicit SDE dynamics for the entire triangular (Gelfand–Tsetlin-type) array — not only for first-coordinate marginals — and prove that the limiting dynamics preserve the half-space Whittaker process.

(d) Degeneration-chain deliverable — uniform Plancherel removal: justify the limit $\tau\downarrow0$ removing the Plancherel component of the specialization uniformly over the parameter regimes, including for multipoint observables, so that the polymer Laplace-transform formulas hold without regime restrictions.

Deliverables (b)–(d) are the structural and rigor components identified by the source paper as the missing support for its asymptotic program; (a) is the multipoint strengthening of that program formulated explicitly by the subsequent literature. A complete answer establishes (a) and settles (b), (c), and (d); resolution of an individual component alone constitutes partial progress rather than completion.

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