  We discuss how, under suitable assumptions, a K\"ahler test configuration
admits a mirror Landau-Ginzburg model, giving a corresponding expression for
the Donaldson-Futaki invariant as a residue pairing. We study the general
behaviour of such mirror formulae under large scaling of the K\"ahler form. We
exploit the observation that this scaling trivially preserves $K$-stability,
but takes the mirror Landau-Ginzburg model to a large complex structure limit.
In certain cases the mirror formulae for the Donaldson-Futaki invariant
simplify in this limit. We focus on a special type of limiting behaviour, when
the Donaldson-Futaki invariant concentrates at a single critical point of the
Landau-Ginzburg potential, and show that this leads to new formulae for the
Donaldson-Futaki invariant in terms of theta functions on the mirror. We
provide a main application, which shows that such limiting behaviour actually
occurs for test configurations in several nontrivial examples, both toric and
non-toric, in the case of slope (in)stability for polarised surfaces.
