In this notes we study the large deviations of the time‐averaged current in the two‐dimensional (2D) Kipnis‐Marchioro‐Presutti model of energy transport when subject to a boundary gradient. We use the tools of hydrodynamic fluctuation theory, supplemented with an appropriate generalization of the additivity principle. As compared to its one‐dimensional counterpart, which amounts to assume that the optimal profiles responsible of a given current fluctuation are time‐independent, the 2D additivity conjecture requires an extra assumption, i.e. that the optimal, divergence‐free current vector field associated to a given fluctuation of the time‐averaged current is in fact constant across the system. Within this context we show that the current distribution exhibits in general non‐Gaussian tails. The ensuing optimal density profile can be either monotone for small current fluctuations, or non‐monotone with a single maximum for large enough current deviations. Furthermore, this optimal profile remains invariant under arbitrary rotations of the current vector, providing a detailed example of the recently introduced Isometric Fluctuation Relation.

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