# 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

1. Macdonald-related dynamics are still open
2. 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}] 
drqd dual $$q$$RSK algorithm [{matveev-petrov15}]
drqe $$q$$RS algorithm [{oconnell-pei13}] [{borodin-petrov13}]
..m. ? ?
d.w. x  x
crsu DLPP with exponential weights [{dieker-warren09}]
crsd - -
crse type-A Pitman's transform [{oconnell03b}]
c.q. x  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 -  -
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.

. In [{pei16}] Local move integrability of the $$q$$RSK was shown, which turns out to be equivalent to the Markov-Doob integrability.

. Not likely to exist because the Whittaker functions are usually defined on continuous space.

. Not likely to exist becasue need to find a continuous version of the $$q$$-Geometric distribution, which is weird.

. 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.