A grove is a spanning forest of a triangular portion of the triangular lattice with a prescribed boundary connectivity. A large random grove exhibits a limit shape i.e. there is a non-random algebraic curve outside which the grove is “frozen”. TK Petersen and D Speyer proved that for the uniform measure on groves, the curve is the inscribed circle. I will talk about extensions of their results to probability measures on groves that are periodic in appropriate coordinates. These measures give interesting algebraic curves with higher genus and cusp singularities as limit shapes, as well as new “gaseous” phases.

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Last updated: 28 May 2024