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比利时vs摩洛哥足彩 ,
university of california san diego

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math 211b - group actions seminar

prof. ben hayes

university of virginia

growth dichotomy for unimodular random rooted trees

abstract:

we show that the growth of a unimodular random rooted tree (t,o) of degree bounded by d always exists, assuming its upper growth passes the critical threshold of the square root of d-1. this complements timar's work who showed the possible nonexistence of growth below this threshold. the proof goes as follows. by benjamini-lyons-schramm, we can realize (t,o) as the cluster of the root for some invariant percolation on the d-regular tree. then we show that for such a percolation, the limiting exponent with which the lazy random walk returns to the cluster of its starting point always exists. we develop a new method to get this, that we call the 2-3-method, as the usual pointwise ergodic theorems do not seem to work here. we then define and prove the cohen-grigorchuk co-growth formula to the invariant percolation setting. this establishes and expresses the growth of the cluster from the limiting exponent, assuming we are above the critical threshold.

host: brandon seward

february 15, 2024

10:00 am

apm 7321

research areas

ergodic theory and dynamical systems

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