The exponential asymptotic behavior of grafting and Teichmüller rays

Gupta showed that for a grafting ray, there is a Teichmüller ray which is asymptotic to the grafting ray. We consider the decrease speed of the Teichmüller distance between the grafting and Teichmüller rays. Based on Gupta’s method, we give a more explicit estimate of the distance and show that given a grafting ray determined by an arational measured lamination, there exists a Teichmüller ray such that the Teichmüller distance between them decreases as a negative exponential function.