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arxiv: 2604.23461 · v1 · submitted 2026-04-25 · 🧮 math.PR · math.CO· math.OC

Scaling limit of Sinkhorn-rescaled Random Matrices via Stability of Static Schr\"odinger Bridges

Danny Duan , Hanbaek Lyu , William Powell This is my paper

Pith reviewed 2026-05-08 07:21 UTC · model grok-4.3

classification 🧮 math.PR math.COmath.OC
keywords Sinkhorn algorithmSchrödinger bridgerandom matricesscaling limitsstability theoryconcentration inequalitiesoptimal transport
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The pith

Sinkhorn-rescaled random matrices with sub-exponential entries converge to the continuous static Schrödinger bridge as dimensions increase.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that for a random matrix with independent sub-exponential entries, its Sinkhorn rescaling concentrates around the rescaling of its mean matrix, both at the level of the Schrödinger potentials and as random measures on the unit square, and supplies explicit non-asymptotic rates for this concentration. As the dimensions tend to infinity, the rescaled random matrix converges to the continuous static Schrödinger bridge fixed by the limiting row and column margins together with a reference density. The argument rests on a new quantitative stability theory for the static Schrödinger bridge that gives Lipschitz continuity in Hellinger distance under reference-measure perturbations, Hölder-1/2 continuity under L1 margin perturbations, and L^∞ stability of the discrete potentials under margin perturbations. These stability bounds translate directly into the concentration and scaling-limit statements for the random-matrix setting.

Core claim

The central claim is that the Sinkhorn rescaling of a random matrix with independent sub-exponential entries concentrates around the rescaling of its mean matrix, both at the level of the Schrödinger potentials and as random measures on the unit square, with explicit non-asymptotic rates; furthermore, as the dimensions tend to infinity, the rescaled random matrix converges to the continuous static Schrödinger bridge determined by the limiting margins and reference density.

What carries the argument

The quantitative stability theory for the static Schrödinger bridge, which establishes Lipschitz continuity in the Hellinger distance under perturbations of the reference measure, Hölder-1/2 continuity under L1 perturbations of the margins, and L^∞ stability of the discrete Schrödinger potentials under margin perturbation.

If this is right

  • The rescaled matrices concentrate around the mean rescaling with explicit rates independent of dimension.
  • Bulk rigidity holds for the empirical spectral distribution of the sample covariance matrix of the rescaled matrix.
  • A central limit theorem holds for the empirical Schrödinger potentials of the rescaled empirical mean.
  • The scaling limit of the rescaled random matrix is exactly the continuous static Schrödinger bridge.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The stability estimates could extend to matrices with other entry distributions or to alternative marginal-enforcement algorithms.
  • The fluctuation results around the bridge may yield limit theorems for eigenvalue statistics in related Gram-matrix models.
  • The discrete-to-continuous bridge connection supplies a template for scaling-limit arguments in other discrete optimal-transport settings.

Load-bearing premise

The matrix entries are independent and sub-exponential, and the limiting row and column margins together with a reference density exist so that the continuous static Schrödinger bridge is well-defined.

What would settle it

A sequence of large random matrices with independent sub-exponential entries whose Sinkhorn potentials or induced measures deviate from those of the mean matrix by more than the stated non-asymptotic rate would falsify the concentration result.

read the original abstract

We analyze the asymptotic behavior and scaling limits of large random matrices rescaled via the Sinkhorn algorithm to match prescribed row and column margins. For a random matrix with independent sub-exponential entries, we show that its Sinkhorn rescaling concentrates around the rescaling of its mean matrix, both at the level of the Schr\"odinger potentials and as random measures on the unit square, with explicit non-asymptotic rates. As the dimensions grow, the rescaled random matrix converges to the continuous static Schr\"odinger bridge (SSB) determined by the limiting margins and reference density. Around this scaling limit we develop a fluctuation theory: bulk rigidity for the empirical spectral distribution of the associated sample covariance matrix, and a central limit theorem for the empirical Schr\"odinger potentials of the rescaled empirical mean. Our analysis is driven by a new quantitative stability theory for the SSB, developed in three forms: Lipschitz continuity in the Hellinger distance under perturbations of the reference measure (kernel stability); H\"older-$1/2$ continuity in the Hellinger distance under $L^1$ perturbations of the margins (margin stability); and $L^\infty$ stability of the discrete Schr\"odinger potentials under margin perturbation (potential stability). Translated to the discrete random-matrix setting, these bounds yield the concentration and scaling-limit results, while a local law for random Gram matrices with a non-uniform variance profile drives the bulk rigidity. Our SSB stability theory may be of independent interest.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript analyzes the asymptotic behavior of large random matrices with independent sub-exponential entries after Sinkhorn rescaling to prescribed row and column margins. It establishes non-asymptotic concentration of the rescaled matrix around the Sinkhorn rescaling of its mean matrix, both at the level of Schrödinger potentials and as random measures on the unit square, with explicit rates. In the scaling limit, the rescaled matrix converges to the continuous static Schrödinger bridge determined by the limiting margins and reference density. The analysis is based on new quantitative stability results for the SSB (Lipschitz continuity in Hellinger distance under reference-measure perturbations, Hölder-1/2 continuity under L¹ margin perturbations, and L^∞ stability of discrete potentials) together with a local law for Gram matrices with non-uniform variance profiles, which additionally yields bulk rigidity for the empirical spectral distribution of the associated sample covariance and a central limit theorem for the empirical Schrödinger potentials.

Significance. If the central claims hold, the work provides a rigorous quantitative bridge between high-dimensional random matrix models and continuous objects from entropic optimal transport. The explicit non-asymptotic rates and the three-form stability theory for static Schrödinger bridges constitute a self-contained technical contribution that may be of independent interest outside random-matrix applications. The extension of local laws to non-uniform variance profiles and the resulting fluctuation theory add depth. The argument is grounded in standard assumptions on sub-exponential entries and the existence of limiting margins and reference density, with no free parameters or circular reductions.

minor comments (3)
  1. Abstract: the abbreviation 'SSB' is introduced without a parenthetical expansion on first use; adding '(static Schrödinger bridge)' would improve immediate readability for readers outside the optimal-transport community.
  2. Introduction (likely §1): the discussion of prior literature on Sinkhorn rescaling for random matrices could usefully cite one or two additional recent works on entropic optimal transport in high dimensions to better situate the novelty of the stability approach.
  3. Notation section: the distinction between discrete Schrödinger potentials and their continuous counterparts is clear in the text but could be reinforced by a short table of symbols at the end of the preliminaries.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, accurate summary of our results, and positive assessment. The recommendation of minor revision is noted; absent any specific major comments or requested changes in the report, we interpret this as an invitation to perform light editorial polishing and will do so in the revised version.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper develops a new quantitative stability theory for the static Schrödinger bridge (Lipschitz in Hellinger under reference perturbations, Hölder-1/2 under margin perturbations, and L^∞ potential stability) and applies it to derive non-asymptotic concentration and scaling limits for Sinkhorn-rescaled random matrices with independent sub-exponential entries. These steps rest on standard concentration tools for sub-exponential variables together with the freshly established stability maps; the assumptions match exactly those required for the continuous SSB to be well-defined. No derivation reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The central claims therefore remain independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of limiting margins and reference density (domain assumption) and on the sub-exponential tail condition for matrix entries (standard in random matrix theory). No free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption Existence of limiting row and column margins and a reference density that determine a well-defined continuous static Schrödinger bridge
    Invoked to state the target scaling limit object.
  • standard math Matrix entries are independent and sub-exponential
    Standard tail assumption used to obtain concentration and local laws.

pith-pipeline@v0.9.0 · 5577 in / 1426 out tokens · 88351 ms · 2026-05-08T07:21:18.562251+00:00 · methodology

discussion (0)

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