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arxiv: 2104.03458 · v3 · submitted 2021-04-08 · 🧮 math.PR · math-ph· math.MP

On the stationary solutions of random polymer models and their zero-temperature limits

David A. Croydon , Makiko Sasada This is my paper

Pith reviewed 2026-05-24 13:20 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords random polymer modelsstationary measureszero-temperature limitsbeta-gamma modelsriver delta modelindependence preservationbijections
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The pith

Stationary measures for zero-temperature random polymer models are obtained by reducing their maps to two basic bijections.

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

The paper establishes stationary measures for zero-temperature versions of the inverse-gamma, gamma, inverse-beta and beta random polymer models. It does so by showing that each zero-temperature map reduces to one of two basic bijections already known from the positive-temperature case, where an independence-preservation property characterizes the stationary distributions. The argument produces explicit descriptions, including new ones for the river delta model, and accounts for atoms that appear because the change of variables degenerates at zero temperature. A reader would care because stationary measures describe the long-run spatial structure of these models, which arise in statistical mechanics and probability.

Core claim

We derive stationary measures for certain zero-temperature random polymer models, which we believe are new in the case of the zero-temperature limit of the beta random polymer (that has been called the river delta model). To do this, we apply techniques developed for understanding the stationary measures of the corresponding positive-temperature random polymer models (and some deterministic integrable systems). More specifically, the article starts with a survey of results for the four basic beta-gamma models, highlighting how the maps underlying the systems in question can each be reduced to one of two basic bijections, and that through an 'independence preservation' property, these bijec

What carries the argument

Two basic bijections equipped with an independence preservation property, applied after reducing each zero-temperature polymer map to one of them.

If this is right

  • Stationary measures are obtained for the river delta model and the other three zero-temperature beta-gamma models.
  • The degeneracy of the change of variables at zero temperature explains the atoms present in some of these measures.
  • Links between the four polymer models, some previously known and some new, follow directly from the shared bijections.
  • The same reduction technique applies to certain deterministic integrable systems at zero temperature.

Where Pith is reading between the lines

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

  • The partial characterization obtained despite degeneracy may be completed by separate arguments that handle the atomic parts directly.
  • The bijection viewpoint could be tested on other zero-temperature models outside the beta-gamma family to see whether the same two maps suffice.
  • Stationary measures derived this way supply candidate initial conditions for studying scaling limits or fluctuation exponents in the zero-temperature setting.

Load-bearing premise

The independence preservation property of the two basic bijections continues to characterize stationary measures even after the change of variables becomes degenerate at zero temperature.

What would settle it

A concrete stationary measure for the river delta model whose marginals or joint laws cannot be recovered from the two bijections would show the reduction does not yield the stationary solutions.

read the original abstract

We derive stationary measures for certain zero-temperature random polymer models, which we believe are new in the case of the zero-temperature limit of the beta random polymer (that has been called the river delta model). To do this, we apply techniques developed for understanding the stationary measures of the corresponding positive-temperature random polymer models (and some deterministic integrable systems). More specifically, the article starts with a survey of results for the four basic beta-gamma models (i.e. the inverse-gamma, gamma, inverse-beta and beta random polymers), highlighting how the maps underlying the systems in question can each be reduced to one of two basic bijections, and that through an `independence preservation' property, these bijections characterise the associated stationary measures. We then derive similar descriptions for the corresponding zero-temperature maps, whereby each is written in terms of one of two bijections. One issue with this picture is that, unlike in the positive-temperature case, the change of variables required is degenerate in general, and so whilst the argument yields stationary solutions, it does not provide a complete characterisation of them. On the other hand, this degeneracy does allow us to explain the appearance of atoms in the stationary measures of certain zero-temperature models. We also derive from our viewpoint various links between random polymer models, some of which recover known results, some of which are novel, and some of which lead to further questions.

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 paper claims to derive stationary measures for certain zero-temperature random polymer models (including the river delta model as a new case) by reducing the underlying maps to two basic bijections from the positive-temperature beta-gamma models and invoking their independence-preservation property. It surveys the inverse-gamma, gamma, inverse-beta and beta polymers, extends the bijection descriptions to the zero-temperature setting, notes that the change-of-variables map degenerates (yielding stationary solutions without a complete characterization), uses this to explain atoms in the measures, and derives various links between the models.

Significance. If the derivations hold, the work provides explicit stationary measures for zero-temperature polymer models and explains the origin of atoms via degeneracy, extending positive-temperature techniques. The reduction to two basic bijections and the independence-preservation property is a methodological strength that produces falsifiable stationary solutions and recovers some known links while identifying novel ones.

minor comments (3)
  1. [Abstract] Abstract: the statement that the argument 'yields stationary solutions' but 'does not provide a complete characterisation' is important; the main text should include an explicit statement (with a concrete example) of what is missing from the characterization after degeneracy.
  2. The survey of the four basic beta-gamma models and their reduction to two bijections should include a table or diagram summarizing which model reduces to which bijection, to make the zero-temperature extension easier to follow.
  3. When deriving links between models, the text should distinguish (e.g., via a remark or subsection) which links recover known results versus which are novel, as the abstract claims both.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No specific major comments were raised in the report, so we have no points to address point-by-point. We will incorporate any minor editorial suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity; techniques applied to new setting with explicit limitations

full rationale

The derivation applies bijections and independence-preservation properties from positive-temperature polymer models to the zero-temperature case. The abstract explicitly flags that the change-of-variables map degenerates at zero temperature, so the argument produces stationary solutions without claiming a complete characterization. This limitation is stated rather than hidden. No load-bearing step reduces by construction to a fitted parameter, self-definition, or unverified self-citation chain; the novelty claim is restricted to specific models where the measures are asserted new. The central claim remains independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; the central claims rest on the independence preservation property of bijections being applicable in the degenerate zero-temperature case, treated as a domain assumption from prior positive-temperature work.

axioms (1)
  • domain assumption Independence preservation property of the two basic bijections characterises the stationary measures
    Invoked to extend positive-temperature results to zero-temperature maps and to explain atoms via degeneracy.

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