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Abstract We define the co-spectral radius of inclusions$${\mathcal S}\leq {\mathcal R}$$of discrete, probability- measure-preserving equivalence relations as the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on$$G/H$$for inclusion$$H\leq G$$of groups. For the proof, we develop a more general version of the 2–3 method we used in another work on the growth of unimodular random rooted trees. We use this method to show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree. We also investigate how the co-spectral radius behaves for hyperfinite relations, and discuss new critical exponents for percolation that can be defined using the co-spectral radius.
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This content will become publicly available on September 2, 2026
An exact formula for the contraction factor of a subdivided Gaussian topological polymer
We consider the radius of gyration of a Gaussian topological polymer G formed by subdividing a graph G' of arbitrary topology (for instance, branched or multicyclic). We give a new exact formula for the expected radius of gyration and contraction factor of G in terms of the number of subdivisions of each edge of G' and a new weighted Kirchhoff index for G'. The formula explains and extends previous results for the contraction factor and Kirchhoff index of subdivided graphs.
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- Award ID(s):
- 2107700
- PAR ID:
- 10637352
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Journal of Physics A: Mathematical and Theoretical
- Volume:
- 58
- Issue:
- 35
- ISSN:
- 1751-8113
- Page Range / eLocation ID:
- 355201
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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