Lozenge Tilings and the Gaussian Free Field on a Cylinder

Ahn, Andrew; Russkikh, Marianna; Peski, Roger Van

doi:10.1007/s00220-022-04491-x

Cited by 4 publications

(4 citation statements)

References 43 publications

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“…It is worth mentioning that Jacobi theta functions have appeared in the related context of periodic Schur processes introduced in [16], used further in e.g. [3], [12], [47,46,48], and their above transformation law has been useful there. We are not aware of any closer connection with the present work, however.…”

Section: Residue Expansionsmentioning

confidence: 99%

“…It is in fact easier to work with random matrices over the p-adic integers 3 , and computations in the p-adic setting are used to find the limit distributions which occur (after taking a product over multiple primes) in the integer setting. A matrix A ∈ Mat N (Q p ) similarly has Smith normal form A = U diag(p SN(A) 1 , .…”

mentioning

confidence: 99%

“…Our usage of 'local' in this work, which instead follows that of the real/complex random matrix theory literature, is unrelated. 3 We encourage a reader less familiar with the p-adic setting to read the first few paragraphs of background in Section 2 before continuing.…”

mentioning

confidence: 99%

“…− t)3 and (1 − t) 2 (1 − t 2 ) respectively to the degree 3 term, also contributing (1 − t)3 and (1 − t) 2 (1 − t 2 ) respectively to the degree 2 term, and 1(8.15) …”

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confidence: 99%

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Limits and fluctuations of p-adic random matrix products

Peski¹

2021

Sel. Math. New Ser.

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We study the distribution of singular numbers of products of certain classes of padic random matrices, as both the matrix size and number of products go to ∞ simultaneously. In this limit, we prove convergence of the local statistics to a new random point configuration on Z, defined explicitly in terms of certain intricate mixed q-series/exponential sums. This object may be viewed as a nontrivial p-adic analogue of the interpolating distributions of Akemann-Burda-Kieburg [6], which generalize the sine and Airy kernels and govern limits of complex matrix products. Our proof uses new Macdonald process computations and holds for matrices with iid additive Haar entries, corners of Haar matrices from GLN (Zp), and the p-adic analogue of Dyson Brownian motion studied in [80]. Contents 1. Introduction 1 2. p-adic matrix background 3. Symmetric function background 4. The limit of the Plancherel/principal Hall-Littlewood measure 5. Residue expansions 6. Tightness and the limiting random variable 7. An indeterminate moment problem 8. Examples of residue formula for L k,t,χ 9. The case of pure α specializations 10. From processes of infinitely many particles to finite matrix bulk limits Appendix A. Parallels with complex matrix products Appendix B. A Hall-Littlewood proof of [80, Theorem 1.2] References

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Section: Residue Expansionsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…− t)3 and (1 − t) 2 (1 − t 2 ) respectively to the degree 3 term, also contributing (1 − t)3 and (1 − t) 2 (1 − t 2 ) respectively to the degree 2 term, and 1(8.15) …”

mentioning

confidence: 99%

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Limits and fluctuations of p-adic random matrix products

Peski¹

2021

Sel. Math. New Ser.

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Two counterparts of the TFK formula for cylinder graphs

Ciucu

2023

Journal of Combinatorial Theory, Series A

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Local limits in p$p$‐adic random matrix theory

Van Peski

2024

Proceedings of London Math Soc

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We study the distribution of singular numbers of products of certain classes of ‐adic random matrices, as both the matrix size and number of products go to simultaneously. In this limit, we prove convergence of the local statistics to a new random point configuration on , defined explicitly in terms of certain intricate mixed ‐series/exponential sums. This object may be viewed as a nontrivial ‐adic analogue of the interpolating distributions of Akemann–Burda–Kieburg, which generalize the sine and Airy kernels and govern limits of complex matrix products. Our proof uses new Macdonald process computations and holds for matrices with iid additive Haar entries, corners of Haar matrices from , and the ‐adic analogue of Dyson Brownian motion studied by the author (https://arxiv.org/pdf/2309.02865).

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Lozenge Tilings and the Gaussian Free Field on a Cylinder

Cited by 4 publications

References 43 publications

Limits and fluctuations of p-adic random matrix products

Limits and fluctuations of p-adic random matrix products

Two counterparts of the TFK formula for cylinder graphs

Local limits in p$p$‐adic random matrix theory

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