Time-like Liouville structure constants and random conformal clusters in statistical models

In the last twenty years there has been an intense effort to describe conformally invariant extended objects in terms of Conformal Field Theories. The fractal dimension of many random paths as well as different geometric exponents controlling, for instance, the reunion probability of an ensemble of self-avoiding walks have been obtained. Despite these great successes, the methods used to study conformally invariant fractals and the comprehension of their hidden mathematical structures remain, in many respects, unsatisfactory. We present here recent numerical results on random fractals which provide motivation and guidance for searching new solutions of Virasoro based field theories. These solutions are expected to be based on time-like Liouville structure constants. Relations to logarithmic aspects of these theories will be discussed.