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<p>We study a Gaussian free field on the hypercube with covariance function related to the transition kernel of a random walk on the hypercube, killed at a geometric random time. An analysis is made of the asymptotic behaviour of the free field as the dimension of the hypercube tends to infinity. We also obtain a result which connects an Ehrenfest urn with geometric killing to a Pólya urn, which is of independent interest.</p>

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Gaussian Free Fields on the Hypercube

  • 期刊论文

  • 发表日期：

    2025年2月6日

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  • Andrea Collevecchio,
  • Robert Griffiths

摘要

We study a Gaussian free field on the hypercube with covariance function related to the transition kernel of a random walk on the hypercube, killed at a geometric random time. An analysis is made of the asymptotic behaviour of the free field as the dimension of the hypercube tends to infinity. We also obtain a result which connects an Ehrenfest urn with geometric killing to a Pólya urn, which is of independent interest.