Fredholm determinantal representations and asymptotics of periodic Macdonald processes at positive charge $u>0$

The periodic Macdonald process (Koshida, 2021) is the $(q,t)$-deformation of the periodic Schur process: a probability measure on cyclic sequences of partitions carrying a charge (finite-temperature) parameter $u$, whose observable moments are computed by free-field (vertex-operator) methods as formal multiple contour integrals of a determinant of a kernel times a universal measure factor $\Delta_{q,t^{-1};u}$. At $u=0$ the measure factor disappears and the generating functions of the observables become Fredholm determinants, recovering the standard Macdonald-process machinery; at the Schur limit $q\to t$ the shift-mixed integrand collapses to a single determinant, recovering the determinantal theory of the periodic Schur process and its finite-temperature asymptotics. At generic $(q,t)$ with $u>0$, however, no Fredholm-determinantal or other integrable-operator representation is known, the rigorous contour realization of the formal integrals exists only for special illustrating cases, and no asymptotics of the periodic $(q,t;u)$ regime have been derived.

Background

Random partitions and symmetric functions supply many exactly solvable stochastic models. A partition $\lambda=(\lambda_1\geq\lambda_2\geq\cdots)$ is a nonincreasing sequence of nonnegative integers; the skew Schur functions $s_{\lambda/\mu}$ and the Macdonald symmetric functions $P_{\lambda/\mu}(\cdot;q,t)$, $Q_{\lambda/\mu}(\cdot;q,t)$ are families of symmetric functions indexed by pairs of nested partitions, evaluated at a specialization $\rho$ (an assignment of values to the power-sum variables $p_n$). The Schur process of Okounkov and Reshetikhin is a probability measure on sequences of interlacing partitions with weights built from $s_{\lambda/\mu}$; it is determinantal, meaning all its correlation functions are minors of a single kernel, which yields Fredholm determinant expressions (determinants of $I+K$ for an operator $K$ acting on a contour) for distribution functions and, through them, asymptotics governed by the Tracy-Widom laws (the universal edge-fluctuation distributions of random matrix theory) and connections to the Kardar-Parisi-Zhang (KPZ) universality class of random growth models. The Macdonald process of Borodin and Corwin is the two-parameter deformation built from $P$ and $Q$; its degenerations $t=0$ (the $q$-Whittaker corner, related to $q$-TASEP and directed polymers) and $q=0$ (the Hall-Littlewood corner, related to the stochastic six-vertex model) describe integrable particle systems and random growth, and the analysis of its observables by Borodin, Corwin, Gorin and Shakirov passes through Fredholm determinants. A half-space variant (Barraquand, Borodin and Corwin) plays the analogous role for models with one open boundary.

Periodicity and charge. Borodin's periodic Schur process makes the sequence of partitions cyclic: for specializations $\rho^+_i,\rho^-_i$ the weight of sequences $\lambda^1,\dots,\lambda^N$ (with inner partitions $\mu^i$, cyclic identification $\lambda^{N+1}=\lambda^1$, and the seam partition $\mu^N$ carrying the charge weight $u^{|\mu^N|}$) is $u^{|\mu^N|}\prod_{i=1}^{N}s_{\lambda^i/\mu^i}(\rho^-_i)s_{\lambda^{i+1}/\mu^i}(\rho^+_i)$, where $|\cdot|$ is total size. The parameter $u\in(0,1)$ is a charge, equivalently finite-temperature or periodicity, parameter: as $u\to0$ the seam partition $\mu^N$ is forced to be empty and the ordinary Schur process is recovered, while for $u>0$ the measures describe cylindric partitions, i.e. plane partitions on a cylinder. Borodin showed that the correlation functions of the periodic process are not determinantal in general but become determinantal after shift-mixing: the point configuration associated with the partitions is translated by an independent integer-valued random variable $n$ with theta-function weight $u^{n^2/2}\zeta^n$. Betea and Bouttier interpreted this as a grand-canonical (finite-temperature) free-fermion ensemble and derived the edge asymptotics, governed by a finite-temperature Airy kernel whose extreme-value statistics interpolate between the Tracy-Widom and Gumbel distributions; Ahn, Russkikh and Van Peski proved Gaussian-free-field fluctuations for cylindric partitions, and Betea and Occelli computed the asymptotic distribution of the peak of cylindric plane partitions as finite-temperature analogues of Bessel and Airy gap probabilities.

The source paper. Koshida's periodic Macdonald process replaces the Schur functions in the cyclic weight by Macdonald ones, $W^{\rho^+,\rho^-}_{q,t;u}(\lambda,\mu)=u^{|\mu^N|}\prod_{i=1}^{N}Q_{\lambda^i/\mu^i}(\rho^-_i;q,t)P_{\lambda^{i+1}/\mu^i}(\rho^+_i;q,t)$. The partition function is computed by a free-field method: the algebra of symmetric functions is identified with a Fock representation of the Heisenberg algebra, the operators diagonalized by Macdonald symmetric functions are realized by vertex operators $\Gamma(\rho)_\pm$ (creation/annihilation-type operators), and the cyclic structure is handled by a trace over the Fock space weighted by the degree operator, which is what produces the charge factor $u^{|\mu^N|}$. The results include a Macdonald-level lift of the MacMahon formula (the classical product formula for the generating function of plane partitions) to cylindric partitions and moment formulas for four series of observables, most importantly $\mathcal{E}_r(\lambda)=e_r(q^\lambda t^{-\rho+1})$, the $r$-th elementary symmetric function evaluated on the specialization with variables $q^{\lambda_i}t^{-i+1}$, $i=1,2,\dots$ (up to the conventional correction term in the power sums), the standard multiplicative encoding of the partition profile; the remaining three series are $\mathcal{E}'_r(\lambda)=e_r(q^{-\lambda}t^{\rho-1})$, $\mathcal{G}_r(\lambda)=g_r(q^\lambda t^{-\rho};q,t)$ and $\mathcal{G}'_r(\lambda)=g_r(q^{-\lambda}t^{\rho};q^{-1},t^{-1})$ (Theorems 5.2, 5.4 and 5.5 of the source), whose moment formulas carry the same universal measure factor. These moments are formal multiple contour integrals of $\det K^{\rho^+,\rho^-}_{q,t;u;\mathcal{E}}(z_i,z_j)$ times a universal measure $\Delta_{q,t^{-1};u}$, a product of cross-Pochhammer factors $(z^{(b)}_j/z^{(a)}_i;u)_\infty$ (where $(a;p)_\infty=\prod_{k\geq0}(1-ap^k)$ is the infinite $q$-Pochhammer symbol) that couple the integration variables across the periodic steps. A shift-mixed extension $\mathbb{P}^{\rho^+,\rho^-}_{q,t;u,\zeta}(\lambda,n)\propto u^{n^2/2}\zeta^n\sum_{\mu}u^{|\lambda|}P_{\lambda/\mu}(\rho^+;q,t)Q_{\lambda/\mu}(\rho^-;q,t)$ is defined (Definition 1.5 of the source, for Macdonald-positive specializations $\rho^\pm$ and an additional parameter $\zeta\in(0,1)$), with moments involving Jacobi theta-function factors and the extended observables $\tilde{\mathcal{E}}_r(\lambda,n)=t^{-rn}\mathcal{E}_r(\lambda)$ (Theorem 1.6), and at the Schur limit $q\to t$ the shift-mixed integrand reduces to a single determinant (Proposition 1.7). Sect. 5.5 of the source gives convergent-contour interpretations of the formal integrals for $N=1$ and exponential specializations $\rho^\pm(p_n)=\sum_s(\alpha^\pm_s)^n$, explicitly as illustrating examples, asserting only that more general cases are handled 'in the analogous way'. The decisive limitation is stated after Corollary 1.4: at $u=0$ the measure $\Delta_{q,t^{-1};u}$ reduces to unity, the integrand becomes a single determinant, and the generating function of $\mathcal{E}_r$ admits a Fredholm determinant expression (recovering the Macdonald-process formulas of Borodin, Corwin, Gorin and Shakirov), but for $u>0$ the integrand is modified by $\Delta_{q,t^{-1};u}$ and no such representation is available.

State of the degenerations. At $q=t$ the analytic program is essentially complete: determinantal shift-mixed kernels, finite-temperature edge asymptotics, and Gaussian-free-field fluctuations. At the $q$-Whittaker corner, Imamura, Mucciconi and Sasamoto proved a restricted Cauchy-sum identity matching the $q$-Whittaker measure to the periodic Schur measure via their Fredholm determinants; He and Wheeler then developed the periodic $q$-Whittaker and Hall-Littlewood processes, proving a $(q,u)$ symmetry of an observable after a random shift, contour-integral formulas, and a matching between periodic Hall-Littlewood observables and the quasi-periodic stochastic six-vertex model, while noting that for the periodic Macdonald process itself the relation to probabilistic models is currently 'only known in the Schur case'. Nothing analogous exists at the full Macdonald level with $u>0$; supplying it is the problem below.

Problem Statement

Objective. Bring the periodic Macdonald process at positive charge to the same standard of exact analysis as its two known degenerations, the Macdonald process at $u=0$ (Fredholm determinants) and the shift-mixed periodic Schur process at $q=t$ (determinantal kernels and finite-temperature asymptotics). The load-bearing question is the one acknowledged in the source: whether genuine Fredholm determinant formulas, or other integrable-operator frameworks, exist for the observables at $u>0$; the rigorous analytic realization and the asymptotic program are its intended continuation.

Part 1 (integrable representation, the core). For $q,t,u\in(0,1)$ with $u>0$ and $q,t$ both generic (not at the Schur, $q$-Whittaker or Hall-Littlewood corners), for Macdonald-positive specializations $\rho^\pm$ and shift-mixing parameter $\zeta\in(0,1)$ as in Definition 1.5 of the source, determine whether the expectation values under the shift-mixed periodic Macdonald measure $\mathbb{P}^{\rho^+,\rho^-}_{q,t;u,\zeta}$ admit a determinantal or integrable-operator representation: a formula for the generating function of the observables $\tilde{\mathcal{E}}_r$ (or of one of the source's other observable series $\mathcal{E}'_r$, $\mathcal{G}_r$, $\mathcal{G}'_r$, whose moment formulas carry the same universal measure factor) in which the universal measure factor $\Delta_{q,t^{-1};u}$ is absorbed into a kernel, in the way the $(z-t^{-1}w)^{-1}$ structure is at $u=0$. The representation is to hold for general admissible specializations $\rho^\pm$ and $\zeta$, not for a single special choice of them. A positive answer consists of constructing such a representation, proving that it equals the measure's expectations, and showing that its formal expansion reproduces the moment formulas of Theorems 1.3 and 1.6 of the source. A negative answer must rule out representations of this kind for the process itself: it must exclude every determinantal or integrable-operator structure whose formal expansion reproduces those moment formulas, since non-existence inside a narrower class that was never able to accommodate the coupling factor $\Delta_{q,t^{-1};u}$ leaves the existence question open.

Part 2 (analytic realization). Provide admissible contour choices with proofs of convergence for the resulting integrals, and for the source's formal moment integrals, valid for general Macdonald-positive specializations, beyond the $N=1$ exponential specializations treated as illustrating examples in the source, so that all moment formulas hold as convergent contour integrals.

Part 3 (asymptotic payoff). Using Parts 1 and 2, derive nontrivial asymptotic laws of the periodic $(q,t;u)$ regime in at least one natural scaling regime (for example, edge fluctuations under Macdonald-Plancherel specializations, the specializations for which the source itself constructs a stationary periodic Macdonald-Plancherel process, its Sect. 4), subject to consistency with the two solved limits wherever the chosen regime admits them: if the chosen regime has a counterpart in the Schur limit $q\to t$, the derived law must degenerate continuously to the corresponding finite-temperature Airy/Gumbel crossover, and if it has a counterpart in the $u\to0$ degeneration, it must reduce to the known Tracy-Widom asymptotics. A regime with no such counterpart (for example a bulk or large-deviation regime) is not required to reproduce these particular edge laws, only to remain consistent with the two degenerations.

The three parts are stages of a single objective and are linked: Part 1 is the core existence question, Part 2 is its rigorous analytic content, and Part 3 is the demonstration that the representation is usable for asymptotics. A positive answer requires all three parts; a rigorous refutation in Part 1 is by itself a complete resolution of the task, and Parts 2 and 3 then do not apply.

References

1. Shinji Koshida, 'Free field theory and observables of periodic Macdonald processes', J. Combin. Theory Ser. A 182 (2021) 105473, DOI 10.1016/j.jcta.2021.105473, arXiv:2001.04607, https://arxiv.org/abs/2001.04607
2. Shinji Koshida, 'Free field approach to the Macdonald process', J. Algebraic Combin. 54 (2021) 223-263, DOI 10.1007/s10801-020-00976-x, arXiv:1905.07087, https://arxiv.org/abs/1905.07087
3. Alexei Borodin, 'Periodic Schur process and cylindric partitions', Duke Math. J. 140 (2007) 391-468, DOI 10.1215/s0012-7094-07-14031-6, arXiv:math/0601019, https://arxiv.org/abs/math/0601019
4. Alexei Borodin, Ivan Corwin, 'Macdonald processes', Probab. Theory Relat. Fields 158 (2014) 225-400, DOI 10.1007/s00440-013-0482-3, arXiv:1111.4408, https://arxiv.org/abs/1111.4408
5. Alexei Borodin, Ivan Corwin, Vadim Gorin, Shamil Shakirov, 'Observables of Macdonald processes', Trans. Amer. Math. Soc. 368 (2016) 1517-1558, DOI 10.1090/tran/6359, arXiv:1306.0659, https://arxiv.org/abs/1306.0659
6. Andrei Okounkov, Nikolai Reshetikhin, 'Correlation function of Schur Process with application to local geometry of a random 3-dimensional Young diagram', J. Amer. Math. Soc. 16 (2003) 581-603, DOI 10.1090/s0894-0347-03-00425-9, arXiv:math/0107056, https://arxiv.org/abs/math/0107056
7. Guillaume Barraquand, Alexei Borodin, Ivan Corwin, 'Half-space Macdonald processes', Forum Math. Pi 8 (2020) e11, DOI 10.1017/fmp.2020.3, arXiv:1802.08210, https://arxiv.org/abs/1802.08210
8. Dan Betea, Jeremie Bouttier, 'The periodic Schur process and free fermions at finite temperature', Math. Phys. Anal. Geom. 22 (2019) 3, DOI 10.1007/s11040-018-9299-8, arXiv:1807.09022, https://arxiv.org/abs/1807.09022
9. Andrew Ahn, Marianna Russkikh, Roger Van Peski, 'Lozenge tilings and the Gaussian free field on a cylinder', Comm. Math. Phys. 396 (2022) 1221-1275, DOI 10.1007/s00220-022-04491-x, arXiv:2105.00551, https://arxiv.org/abs/2105.00551
10. Dan Betea, Alessandra Occelli, 'Peaks of cylindric plane partitions', arXiv:2111.15538 (2021), https://arxiv.org/abs/2111.15538
11. Takashi Imamura, Matteo Mucciconi, Tomohiro Sasamoto, 'Identity between Restricted Cauchy Sums for the $q$-Whittaker and Skew Schur Polynomials', SIGMA 20 (2024) 064, DOI 10.3842/SIGMA.2024.064, arXiv:2106.11913, https://arxiv.org/abs/2106.11913
12. Jimmy He, Michael Wheeler, 'Periodic $q$-Whittaker and Hall-Littlewood processes', Selecta Math. (N.S.) 32 (2026) 11, DOI 10.1007/s00029-025-01113-x, arXiv:2310.03527, https://arxiv.org/abs/2310.03527
13. Jimmy He, Michael Wheeler, 'Free boundary $q$-Whittaker and Hall-Littlewood processes', arXiv:2512.02267 (2025), https://arxiv.org/abs/2512.02267
14. Guillaume Barraquand, 'Integral formulas for two-layer Schur and Whittaker processes', Bull. Soc. Math. France (2026), DOI 10.24033/bsmf.2923, arXiv:2409.08927, https://arxiv.org/abs/2409.08927
15. Sayan Das, Yuchen Liao, Matteo Mucciconi, 'Large deviations for the $q$-deformed polynuclear growth', Ann. Probab. 53 (2025), no. 4, DOI 10.1214/24-AOP1733, arXiv:2307.01179, https://arxiv.org/abs/2307.01179
16. Evgeni Dimitrov, 'KPZ and Airy limits of Hall-Littlewood random plane partitions', Ann. Inst. Henri Poincare Probab. Statist. 54 (2018) 640-693, DOI 10.1214/16-aihp817, arXiv:1602.00727, https://arxiv.org/abs/1602.00727
17. Cesar Cuenca, Matteo Mucciconi, 'The Symplectic Schur Process', Adv. Math. 502 (2026) 111161, DOI 10.1016/j.aim.2026.111161, arXiv:2407.02415, https://arxiv.org/abs/2407.02415
18. S. Ole Warnaar, 'The $A_2$ Andrews-Gordon identities and cylindric partitions', Trans. Amer. Math. Soc. Ser. B 10 (2023) 715-765, DOI 10.1090/btran/147, arXiv:2111.07550, https://arxiv.org/abs/2111.07550
19. Jan Felipe van Diejen, Erdal Emsiz, Ignacio N. Zurrian, 'Affine Pieri rule for periodic Macdonald spherical functions and fusion rings', Adv. Math. 392 (2021) 108027, DOI 10.1016/j.aim.2021.108027, arXiv:2305.01931, https://arxiv.org/abs/2305.01931
20. Takashi Imamura, Matteo Mucciconi, Tomohiro Sasamoto, 'Solvable models in the KPZ class: approach through periodic and free boundary Schur measures', Ann. Probab. 54 (2026), no. 1, DOI 10.1214/25-aop1767, arXiv:2204.08420, https://arxiv.org/abs/2204.08420