{"paper":{"title":"Free field theory and observables of periodic Macdonald processes","arxiv_id":"2001.04607","primary_cat":"math.CO","submitted_at":"2020-01-14T03:27:09+00:00","version_compiled":"v3"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v3","sentences":[{"i":0,"p":0.99,"id":"s0","own":true,"role":"contribution","text":"We propose periodic Macdonald processes as a $(q,t)$-deformation of periodic Schur processes and a periodic analogue of Macdonald processes.","hedge":{"p":0.92,"label":"asserted"},"own_p":0.86,"main_claim_p":0.14,"quantitative":false,"quantitative_p":0.25,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.01,"limitation":0.0,"future_work":0.0,"contribution":0.99,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":false,"role":"background","text":"It is known that, in the theory of stochastic processes related to a family of symmetric functions, the Cauchy-like identity gives an explicit expression of a partition function.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.04,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":0.54,"id":"s2","own":true,"role":"contribution","text":"We compute the partition functions of periodic Macdonald processes relying on the free field realization of the Macdonald theory.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.31,"quantitative":false,"quantitative_p":0.03,"role_probabilities":{"gap":0.0,"method":0.03,"result":0.43,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.54,"interpretation":0.0}},{"i":3,"p":0.84,"id":"s3","own":true,"role":"contribution","text":"We also study several families of observables for periodic Macdonald processes and give formulas of their moments.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.19,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.16,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.84,"interpretation":0.0}},{"i":4,"p":1.0,"id":"s4","own":false,"role":"method","text":"The technical tool is the free field realization of operators that are diagonalized by the Macdonald symmetric functions, where the operators admit expressions by means of vertex operators.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.42,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.03,"role_probabilities":{"gap":0.0,"method":1.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":5,"p":0.95,"id":"s5","own":true,"role":"result","text":"We show that, when we adopt Plancherel specializations, the corresponding periodic Macdonald process is related to a Young diagram-valued periodic continuous process.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.12,"quantitative":false,"quantitative_p":0.12,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.95,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.04,"interpretation":0.01}},{"i":6,"p":0.94,"id":"s6","own":false,"role":"background","text":"It is known that, for periodic Schur processes, determinantal formulas are only available when we extend them to take into account the charge.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.04,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.06,"method":0.0,"result":0.0,"background":0.94,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":7,"p":0.99,"id":"s7","own":true,"role":"contribution","text":"We also consider this kind of shift-mixed periodic Macdonald processes and discuss their 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