Tracy-Widom distribution in quantum XXZ spin chain in the strong anisotropy limit

It has been well known that statistics of current or height of various stochastic models in the Kardar-Parisi-Zhang universality class is described by the Tracy-Widom distributions. In [1], Saenz, Tracy and Widom studied statistics of the position of the left most up-spin in the quantum XXZ spin chain, starting from the domain wall initial condition. They discussed a scaling behavior of the  quantity, mention a possible appearance of the Tracy-Widom distribution but ended with a certain conjecture. An exception was the case of the XX spin chain, with the anisotropy parameter being zero, for which one finds a determinantal formula and gets the Tracy-Widom distribution [2] In our recent work [3] we have studied the so-called Folded XXZ spin chain, which is a strong anisotropy limit, and found the Tracy-Widom distribution for the case of half-alternating initial condition. In this talk we explain this result.