{"paper":{"title":"Fluctuations for the Toda lattice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Fluctuations of currents in the Toda lattice converge to an explicit Gaussian limit under diffusive scaling.","cross_cats":["math.DS"],"primary_cat":"math.PR","authors_text":"Amol Aggarwal, Matthew Nicoletti","submitted_at":"2026-04-15T18:59:53Z","abstract_excerpt":"In this paper we consider the Toda lattice $(\\mathbf{p}(t);\\mathbf{q}(t))$ at thermal equilibrium, meaning that its variables $(p_j)$ and $(e^{q_j-q_{j+1}})$ are independent Gaussian and Gamma random variables, respectively. We show under diffusive scaling that the space-time fluctuations for the model's currents converge to an explicit Gaussian limit. As consequences, we deduce, (i) the scaling limit for the trajectory of a single particle $q_0$ is a Brownian motion; (ii) space-time two-point correlation functions for the model decay inversely with time, with explicit scaling distributions pr"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show under diffusive scaling that the space-time fluctuations for the model's currents converge to an explicit Gaussian limit. As consequences, we deduce (i) the scaling limit for the trajectory of a single particle q_0 is a Brownian motion; (ii) space-time two-point correlation functions decay inversely with time with explicit scaling distributions predicted by Spohn.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The starting assumption that the Toda lattice variables (p_j) and (e^{q_j - q_{j+1}}) are independent Gaussian and Gamma random variables respectively, together with the modeling of the system as a dense collection of quasi-particles whose scattering produces the dressed Lévy-Chentsov field.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Currents in the thermal Toda lattice have space-time fluctuations converging to an explicit Gaussian process under diffusive scaling, implying Brownian motion for particle positions and inverse-time decaying correlations.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Fluctuations of currents in the Toda lattice converge to an explicit Gaussian limit under diffusive scaling.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"6cea5f54e9e7bacefe0e51856ae9208a82f64ea6ebba9375bccaf23ddd1f2623"},"source":{"id":"2604.14346","kind":"arxiv","version":2},"verdict":{"id":"60dee785-9690-40ce-b5c3-fbbe44db9fa2","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T11:53:32.636180Z","strongest_claim":"We show under diffusive scaling that the space-time fluctuations for the model's currents converge to an explicit Gaussian limit. As consequences, we deduce (i) the scaling limit for the trajectory of a single particle q_0 is a Brownian motion; (ii) space-time two-point correlation functions decay inversely with time with explicit scaling distributions predicted by Spohn.","one_line_summary":"Currents in the thermal Toda lattice have space-time fluctuations converging to an explicit Gaussian process under diffusive scaling, implying Brownian motion for particle positions and inverse-time decaying correlations.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The starting assumption that the Toda lattice variables (p_j) and (e^{q_j - q_{j+1}}) are independent Gaussian and Gamma random variables respectively, together with the modeling of the system as a dense collection of quasi-particles whose scattering produces the dressed Lévy-Chentsov field.","pith_extraction_headline":"Fluctuations of currents in the Toda lattice converge to an explicit Gaussian limit under diffusive scaling."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.14346/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}