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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.14346 |
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| _version_ | 1866909054774804480 |
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| author | Aggarwal, Amol Nicoletti, Matthew |
| author_facet | Aggarwal, Amol Nicoletti, Matthew |
| contents | 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 predicted by Doyon (SciPost Phys. 5 (2018), 054) and Spohn (J. Phys. A 53 (2020), 265004). Our starting point is the notion that the Toda lattice can be thought of as a dense collection of many ``quasi-particles'' that interact through scattering. The core of our work is to establish that the full joint scaling limit of the fluctuations for these quasi-particles is given by a Gaussian process, called a dressed Lévy-Chentsov field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14346 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fluctuations for the Toda lattice Aggarwal, Amol Nicoletti, Matthew Probability Dynamical Systems 82C23, 37K10 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 predicted by Doyon (SciPost Phys. 5 (2018), 054) and Spohn (J. Phys. A 53 (2020), 265004). Our starting point is the notion that the Toda lattice can be thought of as a dense collection of many ``quasi-particles'' that interact through scattering. The core of our work is to establish that the full joint scaling limit of the fluctuations for these quasi-particles is given by a Gaussian process, called a dressed Lévy-Chentsov field. |
| title | Fluctuations for the Toda lattice |
| topic | Probability Dynamical Systems 82C23, 37K10 |
| url | https://arxiv.org/abs/2604.14346 |