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Off-Critical Logarithmic Minimal Models

We consider the integrable minimal models ${\cal M}(m,m&#39;;t)$, corresponding to the $φ_{1,3}$ perturbation off-criticality, in the {\it logarithmic limit\,} $m, m&#39;\to\infty$, $m/m&#39;\to p/p&#39;$ where $p, p&#39;$ are coprime and the limit is taken through coprime values of $m,m&#39;$. We view these off-critical minimal models ${\cal M}(m,m&#39;;t)$ 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 ${\cal LM}(p,p&#39;;t)$ corresponding to the $φ_{1,3}$ perturbation of the critical logarithmic minimal models ${\cal LM}(p,p&#39;)$. 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 ${\cal LM}(p,p&#39;)$. 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 $β_{r,s}=(2-α)\,Δ_{r,s}^{p,p&#39;}$ produce all conformal dimensions $Δ_{r,s}^{p,p&#39;}<{(p&#39;-p)(9p-p&#39;)\over 4pp&#39;}$ in the infinitely extended Kac table. The corresponding Kac labels $(r,s)$ satisfy $(p s-p&#39; r)^2< 8p(p&#39;-p)$. The exponent $2-α={p&#39;\over 2(p&#39;-p)}$ is obtained from the logarithmic limit of the free energy giving the conformal dimension $Δ_t={1-α\over 2-α}={2p-p&#39;\over p&#39;}=Δ_{1,3}^{p,p&#39;}$ for the perturbing field $t$. As befits a non-unitary theory, some observables ${\cal O}_{r,s}$ diverge at criticality.