On the discrete side, the uniformly random GT-patterns with fixed top row (or base) as k->\infty are studied in details in arXiv:1202.3901, arXiv:1206.5123, arXiv:1311.5780, arXiv:1604.01110 by two different approaches. The results in the continuous setting can be obtained from the discrete ones by a degeneration (all the methods also survive in this degeneration). There was also one earlier result directly in the continuous setting, which appeared before it was understood that the analysis for the discrete is possible, see arXiv:1105.1272

]]>Good point! I carelessly assumed that there had to be a connection between Schur-Weyl duality and the Weyl group due to the appearance of Weyl’s name in both, but I should have looked into this a bit more carefully.

]]>In fact Schur-Weyl connects S_n to GL_k where n is not equal to k. Instead n is analogous to the sum of the lambdas. So if there existed re-variable symmetric groups they might be relevant with fixed k and real-valued lambda, as the letter a suggested. But I don’t think it is possible to define these groups.

In particular this might not be very related to the Weyl group, as that works only when n=k.

]]>Such a rescaling is done for the symplectic Schur functions by Bisi and Zygouras in https://arxiv.org/abs/1703.07337, in e.g. theorem 4.2 (following O’Connell-Seppalainen-Zygouras for the classical Schur case). The functions are called $ s^{\mathrm{cont}}_\lambda(x) $ for “continuum” version of the Schur functions. The analogue of the HCIZ integral (in the GT form) are given, for Whittaker functions, by identities of Stade and Ishii. Such a rescaling is called there “$0$-temperature limit, in the context of the log-Gamma polymer, so, no, this is not really the limit performed to go from discrete to continuous statistical mechanics models, more to stay on the same model but replace the classical $ (+, \cdot) $ algebra by the max+ algebra (the tropical version of the statistical model). One stays on the same discrete grid, but the log-gamma partition function becomes the last passage percolation time (see proposition 4.1).

For a nice review on these continuous Schur functions and their links with stochastic integrable models, see the survey by O’Connell https://arxiv.org/pdf/1201.4849.pdf (it is named $ J_\lambda(x) $ here).

]]>Professor Tao takes advice from the KKK tsk tsk :P.

]]>*[That worked, thanks! -T.]*

*[Corrected, thanks – T.]*

Good question! I think there should be something on the continuous side, namely finding the limiting shape of a Gelfand-Tsetlin pattern chosen uniformly at random in the limit , assuming the distribution of the base converges to a limit. This shape should be consistent with the free convolution that shows up in the limiting distribution of sums of random Hermitian matrices with spectrum converging to a given limit. Curiously I could not find any literature on this though – there is certainly literature on random standard Young tableau or random Young shapes with the Plancherel distribution, but this isn’t quite the right setup for what is needed here. I might think about it further.

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