Towards Three-Dimensional Conformal Probability

Abstract

In this outline of a program, based on rigorous renormalization group theory, we introduce new definitions which allow one to formulate precise mathematical conjectures related to conformal invariance as studied by physicists in the area known as higher-dimensional conformal bootstrap which has developed at a breathtaking pace over the last few years. We also explore a second theme, intimately tied to conformal invariance for random distributions, which can be construed as a search for very general first and secondquantized Kolmogorov-Chentsov Theorems. First-quantized refers to regularity of sample paths. Secondquantized refers to regularity of generalized functionals or Hida distributions and relates to the operator product expansion. We formulate this program in both the Archimedean and p-adic situations. Indeed, the study of conformal field theory and its connections with probability provides a golden opportunity where padic analysis can lead the way towards a better understanding of open problems in the Archimedean setting. Finally, we present a summary of progress made on a p-adic hierarchical model and point out possible connections to number theory. Parts of this article were presented in author's talk at the 6th International Conference on p-adic Mathematical Physics and its Applications, Mexico 2017.conformal invariance (CI). The simplest example is provided by the time-inversion invariance of Brownian motion tB 1/t d = B t proved by Paul Lévy in [122]. Much deeper is the analogous invariance under conformal transformation in the 2D space of "time parameters" for the Ising model critical scaling limit [55,59] which represents the culmination of major effort led by Schramm, Smirnov and others. Define the lattice couplings

Critical Exponents for Long-Range $${O(n)}$$ O ( n ) Models Below the Upper Critical Dimension

“…(9.14) We also need a lower bound on A N for n = 4. By (5.5), (5.10), and Lemmas 5.4-5.5, apart from a bounded number of scales near 0 and near j m , for j ≤ j m we have δ j [w (2) ] ≍ L ǫj . By restricting the sum in (9.10) to avoid those few scales, the desired lower bound on A N follows similarly.…”

“…The result is that B N also obeys the upper bound (9.14), and consequently so does −u ′′ N . For a lower bound on B N for n = 4, as in (9.10), but now taking into account the exponential decay above the mass scale, there is a z > 0 such that 15) where the (nonnegative) δ[w (2) ] term has been discarded in the lower bound. This is of the same order in j m as the upper bound on A N , but it contains an additional factors, so it cannot spoil the lower bound provided by A N .…”

“…The long-range model has also recently been studied in connection with conformal invariance of the critical theory for d = d c − ǫ with ǫ small [2,82]. From a mathematical point of view, the long-range O(n) model has the advantage that it can be defined in integer dimension d with α = 1 2 (d+ǫ) chosen so that d is just slightly below the upper critical dimension: d = d c − ǫ = 2α − ǫ.…”

“…None of these papers address the computation of critical exponents. Critical correlation functions were studied in a hierarchical version of the model in [48,49], and the recent paper [3] carries out a computation of critical exponents in a different hierarchical setting; see also [2].…”

A Second-Quantized Kolmogorov–Chentsov Theorem via the Operator Product Expansion

“…For completeness, we end §6 with a proof of conformal invariance for the FMFF example. This was previously sketched in [5] to which we refer the reader for more background. When d = 2 and for some smooth volume cut-off function g(x), our result also shows that…”