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What are some of your favorite examples of enumerative problems whose answer ended up being (related to) a solution to one of the Painlevé equations?

I have seen examples from enumeration of classes of graphs, to Hodge integrals and Hurwitz numbers and I'm sure there are many more out there. Bonus points if there is an explanation/heuristic that demystifies or explains why one should expect coefficients of a Painlevé transcendent to appear.

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    $\begingroup$ In random matrix theory, the Tracy-Widom distribution, which is related to Painlevé II, is quite fundamental. There is a close connection between this random matrix theory stuff and limit shapes for combinatorial objects like random Young diagrams according to Plancherel measure and random lozenge tilings. The Airy function, which I think is also closely related to the Painlevé equations, features prominently as well. $\endgroup$ – Sam Hopkins Aug 30 at 19:11
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    $\begingroup$ Are you asking only about enumeration problems? The most famous appearance of solutions of PVI on my opinion are Hitchin's solutions of equations of general relativity. But this is not an enumeration problem. $\endgroup$ – Alexandre Eremenko Aug 31 at 11:30
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    $\begingroup$ @AlexandreEremenko: I would be happy to hear about all surprising appearances. The restriction to enumeration style questions was simply an attempt at keeping the question somewhat focused, but maybe it is worth it to remove that restriction. $\endgroup$ – Gjergji Zaimi Aug 31 at 17:33
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The OP likely has surprising appearances in the mathematical literature in mind, but perhaps the physical observation of a Painléve transcendent is noteworthy?

In Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals (2001) the growth of a liquid crystal cluster is observed in time. The time-dependent probability distribution $\rho$ of the radius $R$ follows (upon rescaling) the solution of the Painléve II equation (dashed curve in the right plot). The origin of the correspondence is described by Craig Tracy in Integrable Probability and the Role of Painlevé Functions.

I am not aware of any other real-world observation of a Painlevé function.

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    $\begingroup$ As the linked slides explain, this is closely related to the random matrix theory connection I mentioned in a comment. $\endgroup$ – Sam Hopkins Sep 1 at 23:36
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I have seen Painlevé transcendents appear a few times in the context of field theory. For example, they appear in model solutions of Hitchin's equations for Higgs fields. (See [MSWW16] and references therein.)

I have also discovered the Hastings-McLeod solution in a certain rescaling limit of the Ginzburg-Landau equations. (Unfortunately this is unpublished.)

Both instances occurred in the context of radially-equivariant solutions to a non-linear elliptic PDE in two dimensions. The resulting ODE is unstable, and there is a distinguished solution which is well-behaved at both $r=0$ and $r=\infty$.

If you have a reasonably-natural nonlinear ODE which is not Painléve, you might be able to add some parameters, take a limit, get lucky, and find something which is Painléve. I also have a vague intuition that their appearance is related to their nice asymptotics.

[MSWW16] Mazzeo, Rafe; Swoboda, Jan; Weiss, Hartmut; Witt, Frederik, Ends of the moduli space of Higgs bundles, Duke Math. J. 165, No. 12, 2227-2271 (2016). ZBL1352.53018.

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