SIGMA 22 (2026), 069, 31 pages      arXiv:2601.06893      https://doi.org/10.3842/SIGMA.2026.069

Absorption Times for Discrete Whittaker Processes and Non-Intersecting Brownian Bridges

Neil O'Connell
School of Mathematics and Statistics, University College Dublin, Dublin 4, Ireland

Received February 20, 2026, in final form July 09, 2026; Published online July 20, 2026

Abstract
We present evidence for a conjectural relationship between absorption times for discrete Whittaker processes and maximal heights of non-intersecting Brownian bridges.

Key words: Toda chain; Whittaker functions.

pdf (804 kb)   tex (301 kb)  

References

1. Baik J., Rains E.M., Algebraic aspects of increasing subsequences, Duke Math. J. 109 (2001), 1-65, arXiv:math.CO/9905083.
2. Biane P., Pitman J., Yor M., Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions, Bull. Amer. Math. Soc. (N.S.) 38 (2001), 435-465, arXiv:math.PR/9912170.
3. Bump D., Automorphic forms on $\operatorname{GL}(3,{\mathbb R})$, Lecture Notes in Math., Vol. 1083, Springer, Berlin, 1984.
4. Bump D., Friedberg S., The exterior square automorphic $L$-functions on $\operatorname{GL}(n)$, in Festschrift in Honor of Piatetski-Shapiro, Part II, Israel Math. Conf. Proc., Vol. 3, Weizmann, Jerusalem, 1990, 47-65.
5. Corwin I., Kardar-Parisi-Zhang universality, Notices Amer. Math. Soc. 63 (2016), 230-239, arXiv:1606.06602.
6. Feierl T., The height of watermelons with wall, J. Phys. A 45 (2012), 095003, 26 pages, arXiv:0802.2691.
7. FitzGerald W., Warren J., Point-to-line last passage percolation and the invariant measure of a system of reflecting Brownian motions, Probab. Theory Related Fields 178 (2020), 121-171, arXiv:1904.03253.
8. Forrester P.J., Log-gases and random matrices, Lond. Math. Soc. Monogr. Ser., Vol. 34, Princeton University Press, Princeton, NJ, 2010.
9. Forrester P.J., Majumdar S.N., Schehr G., Non-intersecting Brownian walkers and Yang-Mills theory on the sphere, Nuclear Phys. B 844 (2011), 500-526, arXiv:1009.2362.
10. Friedberg S., Goldfeld D., Mellin transforms of Whittaker functions, Bull. Soc. Math. France 121 (1993), 91-107.
11. Gerasimov A., Kharchev S., Lebedev D., Oblezin S., On a Gauss-Givental representation of quantum Toda chain wave function, Int. Math. Res. Not. 2006 (2006), 96489, 23 pages, arXiv:math.RT/0505310.
12. Hashizume M., Whittaker functions on semisimple Lie groups, Hiroshima Math. J. 12 (1982), 259-293.
13. Ishii T., Stade E., New formulas for Whittaker functions on $\operatorname{GL}(n,\mathbb R)$, J. Funct. Anal. 244 (2007), 289-314.
14. Kobayashi N., Izumi M., Katori M., Maximum distributions of bridges of noncolliding Brownian paths, Phys. Rev. E 78 (2008), 051102, 15 pages, arXiv:0808.3635.
15. Liechty K., Nonintersecting Brownian motions on the half-line and discrete Gaussian orthogonal polynomials, J. Stat. Phys. 147 (2012), 582-622, arXiv:1111.4239.
16. Nguyen G.B., Remenik D., Non-intersecting Brownian bridges and the Laguerre orthogonal ensemble, Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017), 2005-2029, arXiv:1505.01708.
17. O'Connell N., Absorption times for discrete Whittaker processes and non-intersecting Brownian bridges, https://github.com/noconnell12/dwp.
18. O'Connell N., Geometric RSK and the Toda lattice, Illinois J. Math. 57 (2013), 883-918, arXiv:1308.4631.
19. O'Connell N., Discrete Whittaker processes, Probab. Math. Phys. 4 (2023), 965-1002, arXiv:2211.05718.
20. Schehr G., Majumdar S.N., Comtet A., Randon-Furling J., Exact distribution of the maximal height of $p$ vicious walkers, Phys. Rev. Lett. 101 (2008), 150601, 4 pages, arXiv:0807.0522.
21. Stade E., Mellin transforms of $\operatorname{GL}(n,\mathbb R)$ Whittaker functions, Amer. J. Math. 123 (2001), 121-161, arXiv:2001.0004.
22. Stade E., Trinh T., Recurrence relations for Mellin transforms of $\operatorname{GL}(n,\mathbb R)$ Whittaker functions, J. Funct. Anal. 280 (2021), 108808, 25 pages, arXiv:2020.10880.