Topological expansion for the 1-hermitian matrix model correlation functions.

Once the two special quantities are determined, all other corrections to the resolvent correlators are determined by equation (4.38).

“Once these two quantities are determined, all other corrections to the resolvent correlators are determined by:4.38”

The loop equations were studied a long time ago by Migdal, but they were put in their most useful form by Eynard.

“The loop equations themselves were studied a long time ago (Migdal 1975), but they were put in its most useful form by Eynard (2004).”

A direct way to derive the equivalence between JT gravity and a double-scaled matrix integral is through the loop equations or topological recursion relations of the matrix model.

“A very direct way to derive the equivalence between JT gravity and a double-scaled matrix integral is through the so-called loop equations or topological recursion relations of the matrix model, which provide a recursive relation that determines all or equivalently , in the 1/L or expansion.”

This form of the loop equations was derived in Eynard (2004), and is written for a spectrum with support on the positive real line; for more general spectral support the expression is the same but with the residue taken at both endpoints.

“This form of the loop equations was derived in Eynard (2004), and is written for the case of a spectrum with support on the positive real line . For more general spectral support , the expression is the same, but with the residue taken at both endpoints.”