Off-Critical Logarithmic Minimal Models

We consider the integrable minimal models \({\cal M}(m,m';t)\), corresponding to the \(\varphi_{1,3}\) perturbation off-criticality, in the {\it logarithmic limit\,} \(m, m'\to\infty\), \(m/m'\to p/p'\) where \(p, p'\) are coprime and the limit is taken through coprime values of \(m,m'\). We view these off-critical minimal models \({\cal M}(m,m';t)\) as the continuum scaling limit of the Forrester-Baxter Restricted Solid-On-Solid (RSOS) models on the square lattice. Applying Corner Transfer Matrices to the Forrester-Baxter RSOS models in Regime III, we argue that taking first the thermodynamic limit and second the {\it logarithmic limit\,} yields off-critical logarithmic minimal models \({\cal LM}(p,p';t)\) corresponding to the \(\varphi_{1,3}\) perturbation of the critical logarithmic minimal models \({\cal LM}(p,p')\). Specifically, in accord with the Kyoto correspondence principle, we show that the logarithmic limit of the one-dimensional configurational sums yields finitized quasi-rational characters of the Kac representations of the critical logarithmic minimal models \({\cal LM}(p,p')\). We also calculate the logarithmic limit of certain off-critical observables \({\cal O}_{r,s}\) related to One Point Functions and show that the associated critical exponents \(\beta_{r,s}=(2-\alpha)\,\Delta_{r,s}^{p,p'}\) produce all conformal dimensions \(\Delta_{r,s}^{p,p'}