  The study of quantum gravity in the form of the holographic duality has
uncovered and motivated the detailed investigation of various diagnostics of
quantum chaos. One such measure is the operator size distribution, which
characterizes the size of the support region of an operator and its evolution
under Heisenberg evolution. In this work, we examine the role of the operator
size distribution in holographic duality for the Sachdev-Ye-Kitaev (SYK) model.
Using an explicit construction of AdS$_2$ bulk fermion operators in a putative
dual of the low temperature SYK model, we study the operator size distribution
of the boundary and bulk fermions. Our result provides a direct derivation of
the relationship between (effective) operator size of both the boundary and
bulk fermions and bulk $\text{SL}(2; \mathbb{R})$ generators.
