Markov-Doob integrable dynamics on triangular arrays
markov_doob_integrability
We give a classification of dynamics on the triangular array that are Markov-Doob integrable here.
We consider four parameters with their possible values in braces:
- Space: The triangular array may be on a discrete space (d) e.g. \(\pint\), or continuous space (c) e.g. \(\real_+\).
- Type: The dynamics might be RSK-type (r), which is weight preserving, or Warren-type (w).
- Function: The special harmonic function \(h\) may be the Schur polynomials (s), the \(q\)-Whittaker functions (q), the Macdonald polynomials (m), or the Whittaker functions (w).
- Special: And the functions may have specialisations, including the usual specialisation / usual-spec (u), the dual specialisation / dual-spec (d), or the exponential specialisation / exp-spec (e).
So in total the possibile dynamics are quadruples in {d, c} x {r, w} x {s, q, m, w} x {u, d, e}, ranging 48 possibilities! We write these quadruples in shorthand. E.g. crwe stands for Continuous-time RSK-type \(h\)-being-Whittaker-functions with Exponential-specialisations dynamics. We also use "." as a wildcard.
Remark 1. The RSK-type dynamics can further split into row- and column- insertions. However for now we do not consider them separately, because they are dual to each other in a rather trivial sense in many cases
Remark 1.5. Both the Schur and \(q\)-Whittaker functions are special cases of the Macdonald polynomials, but we do not neglect the former two because
- Macdonald-related dynamics are still open
- The Schur cases are less messy than the \(q\)-case, and have their own merits. This is also the reason we do not neglect Whittaker-related dynamics even though they are limits of the \(q\)-cases.
Let us list all the combinations (?: open; x: likely nonexistent; -: may be out there somewhere):
SpaceTypeFunctionSpecial | name of the dynamics | Markov-Doob integrability shown in... |
---|---|---|
drsu | RSK correspondence | [{oconnell03a}] |
drsd | dual RSK correspondence | |
drse | RS correspondence | |
drqu | \(q\)RSK algorithm | [{matveev-petrov15}] [{pei16}] [0] |
drqd | dual \(q\)RSK algorithm | [{matveev-petrov15}] |
drqe | \(q\)RS algorithm | [{oconnell-pei13}] [{borodin-petrov13}] |
..m. | ? | ? |
d.w. | x [1] | x |
crsu | DLPP with exponential weights | [{dieker-warren09}] |
crsd | - | - |
crse | type-A Pitman's transform | [{oconnell03b}] |
c.q. | x [2] | x |
crwu | gRSK algorithm | [{corwin-oconnell-seppalainen-zygouras14}] |
crwd | - | - |
crwe | semi-discrete polymer | [{oconnell12a}] |
dwsu | push-block | [{warren-windridge09}] |
dwsd | - | - |
dwse | push-block | [{warren-windridge09}] [{borodin-ferrari14}] |
dwqu | usual-spec \(q\)push-block | [{matveev-petrov15}] |
dwqd | dual-spec \(q\)push-block | |
dwqe | exp-spec \(q\)push-block | [{borodin-corwin14}] |
cwsu | - [3] | - |
cwsd | - | - |
cwse | Warren process | [{warren07}] |
Remark 2. The Markov-Doob integrability given Space, Function, and Special has the same statement with the same transition kernels of the shape. Therefore for any x, y and z, the combination xryz and xwyz yield the same Corollary there.
[0]. In [{pei16}] Local move integrability of the \(q\)RSK was shown, which turns out to be equivalent to the Markov-Doob integrability.
[1]. Not likely to exist because the Whittaker functions are usually defined on continuous space.
[2]. Not likely to exist becasue need to find a continuous version of the \(q\)-Geometric distribution, which is weird.
[3]. TODO: Is this the Laguerre Warren process in [{sun16}]?
References
- [borodin-corwin14] Macdonald processes, , Probability Theory and Related Fields, Vol. 158, No. 1-2, p.225–400 2014.
- [borodin-ferrari14] Anisotropic growth of random surfaces in 2+ 1 dimensions, , Communications in Mathematical Physics, Vol. 325, No. 2, p.603–684 2014.
- [borodin-petrov13] Nearest neighbor Markov dynamics on Macdonald processes, , arXiv:1305.5501 [math-ph], arXiv: 1305.5501, may 2013. [ link ]
- [corwin-oconnell-seppalainen-zygouras14] Tropical combinatorics and Whittaker functions, , Duke Mathematical Journal, Vol. 163, No. 3, p.513–563 2014. [ link ]
- [dieker-warren09] On the largest-eigenvalue process for generalized wishart random matrices, , Alea, Vol. 6, p.369–376 2009.
- [matveev-petrov15] q-randomized Robinson-Schensted-Knuth correspondences and random polymers, , 2016.
- [oconnell-pei13] A $q$-weighted version of the Robinson-Schensted algorithm, , Electronic Journal of Probability, Vol. 18, No. 0, oct 2013. [ link ]
- [oconnell03a] Conditioned random walks and the RSK correspondence, , Journal of Physics A: Mathematical and General, Vol. 36, No. 12, p.3049–3066, March 2003.
- [oconnell03b] A path-transformation for random walks and the Robinson-Schensted correspondence, , Transactions of American Mathematical Society, Vol. 355, No. 9, p.3669–3697 2003.
- [oconnell12a] Directed Polymers and the Quantum Toda Lattice, , The Annals of Probability, Vol. 40, No. 2, p.437–458 2012.
- [pei16] A $q$-Robinson-Schensted-Knuth Algorithm and a $q$-polymer, , 2016.
- [sun16] Laguerre and Jacobi analogues of the Warren process, , arXiv preprint arXiv:1610.01635 2016.
- [warren-windridge09] Some examples of dynamics for Gelfand-Tsetlin patterns, , Electron. J. Probab, Vol. 14, p.1745–1769 2009.
- [warren07] Dyson's Brownian motions, intertwining and interlacing, , Electronic Journal of Probability, Vol. 12, p.573–590 2007.