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We study the return probability and its imaginary ($\tau$) time continuationafter a quench from a domain wall initial state in the XXZ spin chain, focusingmainly on the region with anisotropy $|\Delta|< 1$. We establish exact Fredholmdeterminant formulas for those, by exploiting a connection to the six vertexmodel with domain wall boundary conditions. In imaginary time, we find theexpected scaling for a partition function of a statistical mechanical model ofarea proportional to $\tau^2$, which reflects the fact that the model exhibitsthe limit shape phenomenon. In real time, we observe that in the region$|\Delta|<1$ the decay for large times $t$ is nowhere continuous as a functionof anisotropy: it is either gaussian at root of unity or exponential otherwise.As an aside, we also determine that the front moves as $x_{\rmf}(t)=t\sqrt{1-\Delta^2}$, by analytic continuation of known arctic curves inthe six vertex model. Exactly at $|\Delta|=1$, we find the return probabilitydecays as $e^{-\zeta(3/2) \sqrt{t/\pi}}t^{1/2}O(1)$. It is argued that thisresult provides an upper bound on spin transport. In particular, it suggeststhat transport should be diffusive at the isotropic point for this quench.

Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain