Recent zbMATH articles in MSC 60B15
https://zbmath.org/atom/cc/60B15
2024-03-13T18:33:02.981707Z
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Bimonoidal structure of probability monads
https://zbmath.org/1528.18015
2024-03-13T18:33:02.981707Z
"Fritz, Tobias"
https://zbmath.org/authors/?q=ai:fritz.tobias
"Perrone, Paolo"
https://zbmath.org/authors/?q=ai:perrone.paolo
Summary: We give a conceptual treatment of the notion of joints, marginals, and independence in the setting of categorical probability. This is achieved by endowing the usual probability monads (like the Giry monad) with a monoidal and an opmonoidal structure, mutually compatible (i.e. a bimonoidal structure). If the underlying monoidal category is cartesian monoidal, a bimonoidal structure is given uniquely by a commutative strength. However, if the underlying monoidal category is not cartesian monoidal, a strength is not enough to guarantee all the desired properties of joints and marginals. A bimonoidal structure is then the correct requirement for the more general case.
We explain the theory and the operational interpretation, with the help of the graphical calculus for monoidal categories. We give a definition of stochastic independence based on the bimonoidal structure, compatible with the intuition and with other approaches in the literature for cartesian monoidal categories. We then show as an example that the Kantorovich monad on the category of complete metric spaces is a bimonoidal monad for a non-cartesian monoidal structure.
For the entire collection see [Zbl 1411.68020].
Weighted cogrowth formula for free groups
https://zbmath.org/1528.20030
2024-03-13T18:33:02.981707Z
"Jaerisch, Johannes"
https://zbmath.org/authors/?q=ai:jaerisch.johannes
"Matsuzaki, Katsuhiko"
https://zbmath.org/authors/?q=ai:matsuzaki.katsuhiko
Summary: We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group \(\text{Cay}(F_n)\) by an arbitrary subgroup \(G\) of \(F_n\). Our main result, which generalizes Grigorchuk's cogrowth formula to variable edge lengths, provides a formula relating the bottom of the spectrum of weighted Laplacian on \(G \backslash \text{Cay}(F_n)\) to the Poincaré exponent of \(G\). Our main tool is the Patterson-Sullivan theory for Cayley graphs with variable edge lengths.