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Pointwise multiplication of random Schwartz distributions with Wilson's operator product expansion

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SRQW02 - Quantum field theory, renormalisation and stochastic partial differential equations

I will present a general theorem for the multiplication of random distributions which is similar in spirit to the construction of local Wick powers of a Gaussian field. However, this theorem is much more general in scope and applies to non-Gaussian measures, even without translation invariance and in the presence of anomalous scaling, provided the random fields involved are less singular than white noise. Conjecturally, the construction of the energy field of the 3D Ising scaling limit as a square of the spin field should fall within the purview of the theorem. Our construction involves multiplying mollified distributions followed by suitable additive and multiplicative renormalizations before a proof of almost-sure convergence when the mollification is removed. The main tools for the proof are combinatorial estimates on moments. The main hypothesis for the theorem is Wilson's OPE with precise quantitative bounds for pointwise correlations at noncoinciding points. I will also explain how the theorem works on the example of a simple conformal field theory of mean field type, namely, the fractional Gaussian field.

This talk is part of the Isaac Newton Institute Seminar Series series.

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