Pith Number

pith:IOJY4PHQ

pith:2026:IOJY4PHQ56CKLWILDXGUC2SSXO

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Fluctuations for the Toda lattice

Amol Aggarwal, Matthew Nicoletti

Fluctuations of currents in the Toda lattice converge to an explicit Gaussian limit under diffusive scaling.

arxiv:2604.14346 v2 · 2026-04-15 · math.PR · math.DS

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

Receipt and verification

First computed	2026-05-20T01:05:13.370286Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

43938e3cf0ef84a5d90b1dcd416a52bb9e80ba4de34536671191faee9cc327d2

Aliases

arxiv: 2604.14346 · arxiv_version: 2604.14346v2 · doi: 10.48550/arxiv.2604.14346 · pith_short_12: IOJY4PHQ56CK · pith_short_16: IOJY4PHQ56CKLWIL · pith_short_8: IOJY4PHQ

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/IOJY4PHQ56CKLWILDXGUC2SSXO \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 43938e3cf0ef84a5d90b1dcd416a52bb9e80ba4de34536671191faee9cc327d2

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "5c8d1f24b8b35afc022ff7425ea85efb850b05f7212f5689650a3051a8f8352b",
    "cross_cats_sorted": [
      "math.DS"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-04-15T18:59:53Z",
    "title_canon_sha256": "fe5dffff0807c616588eea29d55d64b6f71bb4ac6d79a7fa73dce4abc52aba10"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.14346",
    "kind": "arxiv",
    "version": 2
  }
}