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How to replace random walk representations in models that don't have one?

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Random walk representations for the correlations of random fields have played a considerable role in the past. For instance, the correlations of gradient models have such a representation if the interactions are given by a convex function of the gradient. Also, correlations for gradient models with local pinning have been analyzed in great details using random walks with traps, see e.g. [1], where very detailed properties of the range of a random walk played a crucial role. In recent years, there had been interest in random fields which don’t admit a direct random walk representation, for instance so-called membrane models. In a recent paper with Alessandra Cipriani and Noemi Kurt [2], we analyzed the decay of correlations for such fields with local pinning which is based on analytic methods. Although the method does not use random walks, it is actually close in spirit to the methods using them. Presently, the results are however much less precise than those obtained for gradient models.

References

[1] Bolthausen, E., and Velenik, Y.: Critical behavior of the massless free field at the depinning transition. Commun. Math. Phys. 223, 161-203 (2001).

[2] Bolthausen, E., Cipriani, A., and Kurt, N.: Exponential Decay of Covari- ances for the Supercritical Membrane Model. Comm. Math. Phys. 2017, doi:10.1007/s00220-017-2886-x.

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