  In this paper, we investigate the second-order normal quantum fluctuation on
the world-sheet of a probe string in the Ba\~nados-Teitelboim-Zanelli (BTZ)
black hole. These fluctuations is treated as the projection of Hawking
radiation on the worldsheet and indeed modify the action growth of the string.
Then in the string field theory/boundary conformal field theory framework, via
the boundary vertex operator we study the correlation function of the
Schr\"odinger functional of excited fields on the world-sheet and further
extract the field's formula. Our study could shed light on the potential
connection between complexity growth and correlation function.
