We develop notions of Rota-Baxter structures and associated Birkhoff factorizations, in the context of min-plus semirings and their thermodynamic deformations, including deformations arising from quantum information measures such as the von Neumann entropy. We consider examples related to Manin's renormalization and computation program, to Markov random fields and to counting functions and zeta functions of algebraic varieties.Comment: 28 pages, LaTe

Neural language models learn, to varying degrees of accuracy, the grammatical properties of natural languages. In this work, we investigate whether there are systematic sources of variation in the language models' accuracy. Focusing on subject-verb agreement and reflexive anaphora, we find that certain nouns are systematically understood better than others, an effect which is robust across grammatical tasks and different language models. Surprisingly, we find that across four orders of magnitude, corpus frequency is unrelated to a noun's performance on grammatical tasks. Finally, we find that a novel noun's grammatical properties can be few-shot learned from various types of training data. The results present a paradox: there should be less variation in grammatical performance than is actually observed.

Abstract. We relate the Eternal Symmetree model of Harlow, Shenker, Stanford, and Susskind to constructions of stochastic processes related to quantum statistical mechanical systems on CuntzKrieger algebras. We extend the eternal inflation model from the Bruhat-Tits tree to quotients by p-adic Schottky groups, again using quantum statistical mechanics on graph algebras.

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