Are there some references about tropical cluster algebras and tropical Laurent phenomenon? I searched on Google but only found one paper: Tropical Plucker functions and their bases .
Thank you very much.
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Let me suggest Lauren Williams and my paper as a starting point:
David Speyer and Lauren Williams, MR 2164397 The tropical totally positive Grassmannian, J. Algebraic Combin. 22 (2005), no. 2, 189--210.
It's earlier than a lot of the other references mentioned, so it isn't as sophisticated but, for that reason, it is a lot more concrete.
There are certainly things related to tropical cluster algebras associated to surfaces. See Fomin, Shapiro, Thurston, https://arxiv.org/pdf/math/0608367, especially section 9.5, and the sequel by Fomin and Thurston, https://arxiv.org/abs/1210.5569, sections 14 and 15.
There is also work by Fock and Goncharov along similar lines, see for example https://arxiv.org/abs/math/0510312. (Note that this is much more accessible than some of their work!)