Based on the rigorous path integral formulation of Liouville Conformal Field Theory initiated by David-Kupiainen-Rhodes-Vargas on the Riemann sphere and David-Rhodes-Vargas on the torus of modulus $\tau$, we give the exact asymptotic behaviour of the 1-point toric correlation function as $\mathrm{ Im}\:\tau\to\infty$. In agreement with formulae predicted within the bootstrap formalism of theoretical physics, our results feature an $(\mathrm{ Im}\:\tau)^{ -3/2}$ decay rate and we identify the derivative of DOZZ formula in the limit.
Categories: math.PR math-ph math.MP