Correlation Lengths for Stochastic Matrix Product States

• URL: http://arxiv.org/abs/2510.07561v1
• Date: Wed, 08 Oct 2025 21:17:43 GMT
• Title: Correlation Lengths for Stochastic Matrix Product States
• Authors: Lubashan Pathirana, Albert H. Werner,
• Abstract summary: We prove the existence of thermodynamic limits of expectations of local observables.<n>In the homogeneous (random translation-invariant) case, for any error tolerance in probability, the two-point function decays exponentially in the distance between the two sites.<n>In the i.i.d. case, the exponential decay still holds with a deterministic rate, with the probability approaching one exponentially fast in the distance.
• Score: 0.8594140167290097
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce a general model of stochastically generated matrix product states (MPS) in which the local tensors share a common distribution and form a strictly stationary sequence, without requiring spatial independence. Under natural conditions on the associated transfer operators, we prove the existence of thermodynamic limits of expectations of local observables and establish almost-sure exponential decay of two-point correlations. In the homogeneous (random translation-invariant) case, for any error tolerance in probability, the two-point function decays exponentially in the distance between the two sites, with a deterministic rate. In the i.i.d. case, the exponential decay still holds with a deterministic rate, with the probability approaching one exponentially fast in the distance. For strictly stationary ensembles with decaying spatial dependence, the correlation decay quantitatively reflects the mixing profile: ($\rho$)-mixing yields polynomial bounds with high probability, while stretched-exponential (resp. exponential) decay in ($\rho$) (resp. ($\beta$)) yields stretched-exponential (resp. exponential) decay of the two-point function, again with correspondingly strong high-probability guarantees. Altogether, the framework unifies and extends recent progress on stationary ergodic and Gaussian translation-invariant ensembles, providing a transfer-operator route to typical correlation decay in random MPS.

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