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Bj\"orner and Ekedahl [Ann. of Math. (2), 170(2): 799-817, 2009] pioneeredthe study of length-counting sequences associated with parabolic lower Bruhatintervals in crystallographic Coxeter groups. In this paper, we study theasymptotic behavior of these sequences in affine Weyl groups. Let $W$ be anaffine Weyl group with corresponding finite Weyl group $W_f$, and let${}^{f}{W}$ denote the set of minimal right $W_f$-coset representatives. Weprove that the length-counting sequences associated with the parabolic lowerBruhat intervals $[e, t_\lambda]^f:=[e, t_\lambda] \cap {}^{f}{W}$ are``asymptotically'' log-concave, where the translation $t_\lambda \in W$corresponds to a dominant coroot lattice element $\lambda$. More precisely, weprove that certain discrete measures constructed from length-counting sequencesconverge weakly to a continuous measure constructed from certain polytope, andwe prove that the density function of the limit is log-concave by applying theclassical Brunn--Minkovski inequality in convex geometry. Our approach relieson the ``dominant lattice formula'', which yields a new bridge between thediscrete nature of Betti numbers of parabolic affine Schubert varieties and thecontinuous nature of the geometry of convex polytopes.

Asymptotic Log-concavity of Dominant Lower Bruhat Intervals via the Brunn--Minkowski Inequality