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	[{matveev-petrov15}]
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, Alexei Borodin, Ivan Corwin, 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, Alexei Borodin, Patrik L Ferrari, Communications in Mathematical Physics, Vol. 325, No. 2, p.603–684 2014.
[borodin-petrov13] Nearest neighbor Markov dynamics on Macdonald processes, Alexei Borodin, Leonid Petrov, arXiv:1305.5501 [math-ph], arXiv: 1305.5501, may 2013. [ link ]
[corwin-oconnell-seppalainen-zygouras14] Tropical combinatorics and Whittaker functions, Ivan Corwin, Neil O'Connell, Timo Seppäläinen, Nikolaos Zygouras, Duke Mathematical Journal, Vol. 163, No. 3, p.513–563 2014. [ link ]
[dieker-warren09] On the largest-eigenvalue process for generalized wishart random matrices, AB Dieker, J Warren, Alea, Vol. 6, p.369–376 2009.
[matveev-petrov15] q-randomized Robinson-Schensted-Knuth correspondences and random polymers, Konstantin Matveev, Leonid Petrov, 2016.
[oconnell-pei13] A $q$-weighted version of the Robinson-Schensted algorithm, Neil O'Connell, Yuchen Pei, Electronic Journal of Probability, Vol. 18, No. 0, oct 2013. [ link ]
[oconnell03a] Conditioned random walks and the RSK correspondence, N. O'Connell, 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, Neil O'Connell, Transactions of American Mathematical Society, Vol. 355, No. 9, p.3669–3697 2003.
[oconnell12a] Directed Polymers and the Quantum Toda Lattice, Neil O'Connell, The Annals of Probability, Vol. 40, No. 2, p.437–458 2012.
[pei16] A $q$-Robinson-Schensted-Knuth Algorithm and a $q$-polymer, Yuchen Pei, 2016.
[sun16] Laguerre and Jacobi analogues of the Warren process, Yi Sun, arXiv preprint arXiv:1610.01635 2016.
[warren-windridge09] Some examples of dynamics for Gelfand-Tsetlin patterns, Jon Warren, Peter Windridge, Electron. J. Probab, Vol. 14, p.1745–1769 2009.
[warren07] Dyson's Brownian motions, intertwining and interlacing, Jon Warren, Electronic Journal of Probability, Vol. 12, p.573–590 2007.