For the critical 2D stochastic heat flow, the minimal number of ε-balls carrying almost all mass is ε^{-2}(log 1/ε)^{-1/2+o(1)}, with the complementary sparsity bound holding simultaneously.
Basic properties of critical lognormal multiplicative chaos.Annals of Probability, 43(5):2205–2249, 2015
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Logarithmic intermittency of the critical 2D SHF
For the critical 2D stochastic heat flow, the minimal number of ε-balls carrying almost all mass is ε^{-2}(log 1/ε)^{-1/2+o(1)}, with the complementary sparsity bound holding simultaneously.