Stochastic Quantization of the Φ33-Model

We study the construction of the Φˆ3/3 -measure and complete the program on the (non-)construction of the focusing Gibbs measures, initiated by Lebowitz, Rose, and Speer [J. Statist. Phys. 50 (1988), no. 3-4, 657-687]. This problem turns out to be critical, exhibiting the following phase transition. In the weakly nonlinear regime, we prove normalizability of the ˆΦˆ3/3 -measure and show that it is singular with respect to the massive Gaussian free field. Moreover, we show that there exists a shifted measure with respect to which the Φˆ3/3 -measure is absolutely continuous. In the strongly nonlinear regime, by further developing the machinery introduced by the authors, we establish non-normalizability of the ˆΦˆ3/3 -measure. Due to the singularity of the ˆΦˆ3/3 -measure with respect to the massive Gaussian free field, this non-normalizability part poses a particular challenge as compared to our previous works. In order to overcome this issue, we first construct a -finite version of the ˆΦˆ3/3 -measure and show that this measure is not normalizable. Furthermore, we prove that the truncated ˆΦˆ3/3. -measures have no weak limit in a natural space, even up to a subsequence.

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