Non-commutative probability insights into the double-scaling limit SYK model with constant perturbations: moments, cumulants and q -independence
Shuang Wu
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Source: Crossref
Published: Jul 29, 2024
DOI: 10.1088/1751-8121/ad65a6
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Abstract Extending the results of Wu (2022 J. Phys. A 55 415207), we study the double-scaling limit Sachdev–Ye–Kitaev model with an additional diagonal matrix with a fixed number c of nonzero constant entries θ . This constant diagonal term can be rewritten in terms of Majorana fermion products. Its specific formula depends on the value of c . We find exact expressions for the moments of this model. More importantly, by proposing a moment-cumulant relation, we reinterpret the effect of introducing a constant term in the context of non-commutative probability theory. This gives rise to a q ~ dependent mixture of independences within the moment formula. The parameter q ~ , derived from the q -Ornstein–Uhlenbeck ( q -OU) process, controls this transformation. It interpolates between classical independence ( q ~ = 1 ) and Boolean independence ( q ~ = 0 ). The underlying combinatorial structures of this model provide the non-commutative probability connections. Additionally, we explore the potential relation between these connections and their gravitational path integral counterparts.
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