Regularity of joint-meet ideals of distributive lattices

Let $L$ be a distributive lattice and $R(L)$ the associated Hibi ring. We compute $\text{reg}R(L)$ when $L$ is a planar lattice and give a lower bound for $\text{reg}R(L)$ when $L$ is non-planar, in terms of the combinatorial data of $L\mathrm{.}$ As a consequence, we characterize the distributive lattices $L$ for which the associated Hibi ring has a linear resolution.