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arxiv: 2605.31300 · v1 · pith:HTHOKBTAnew · submitted 2026-05-29 · 🧮 math.PR

Cone-Additive Functions for Random Walks on Free Products of Graphs

Lorenz A. Gilch , Hanna Oppelmayer This is my paper

Pith reviewed 2026-06-28 21:07 UTC · model grok-4.3

classification 🧮 math.PR
keywords cone-additive functionsrandom walksfree products of graphslimit theoremsentropydriftlamplighter walksasymptotic range
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The pith

Cone-additive functions on free products of graphs satisfy a limit theorem that covers entropy, drift, and lamplighter distances.

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

The paper defines cone-additive functions for random walks on free products of countable graphs and shows these functions obey a limit theorem under mild assumptions. Cone-additivity holds for standard quantities including entropy, asymptotic range, and drift. The key feature is a separation by space on the product structure, which differs from the usual time-based subadditivity and yields new limit theorems for travelling salesman problems on lamplighter walks, edge weight functions, and the range of the r-th visit.

Core claim

Cone-additive functions are defined for random walks on free products of countable graphs. These functions satisfy a limit theorem under mild assumptions. Cone-additivity is present in entropy, asymptotic range, and drift. The property acts as a separation by space, in contrast to sub-additivity as a separation by time, and this allows new limit theorems for travelling salesman problems on lamplighter random walks on free products, for weight functions on edges, and for the range of the r-th visit.

What carries the argument

Cone-additive functions, which encode separation by space on the free product structure rather than separation by time.

If this is right

  • Entropy, asymptotic range, and drift on these graphs are cone-additive and therefore satisfy the limit theorem.
  • Distance functions arising from lamplighter random walks on free products obey new limit theorems.
  • Weight functions on edges of the free product graphs satisfy the limit theorem.
  • The range of the r-th visit to a vertex satisfies the limit theorem.

Where Pith is reading between the lines

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

  • The spatial separation idea could apply to other inhomogeneous random walk settings beyond free products.
  • It offers an alternative route to convergence results that avoids reliance on temporal subadditivity.
  • The travelling salesman application suggests possible extensions to other optimization functionals on the same graphs.

Load-bearing premise

The limit theorem holds only under unspecified mild assumptions on the random walks or the functions themselves.

What would settle it

A concrete counterexample consisting of a cone-additive function on a free product of graphs for which the claimed limit theorem fails even when the mild assumptions are met.

read the original abstract

We define cone-additive functions for random walks on free products of countable graphs. These functions satisfy a limit theorem under mild assumptions. In fact, cone-additivity is present in several well-studied notions, like entropy, asymptotic range and drift. Cone-additivity can be seen as a separation property by space -- a quite different perspective than the well-studied concept of sub-additivity in the context of free products of groups, which is a separation by time. In our inhomogeneous setting of free products of graphs, this separation by space allows us to deduce new limit theorems for travelling salesman problems (that is, distance functions of lamplighter random walks on free products), for weight functions on edges and the range of the $r$-th visit.

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

3 major / 2 minor

Summary. The paper defines cone-additive functions on random walks on free products of countable graphs via a separation-by-space property. It establishes that these functions obey a limit theorem under mild assumptions, shows that entropy, asymptotic range and drift are cone-additive, and derives new limit theorems for the travelling salesman problem on lamplighter walks, edge-weight functions, and the range of the r-th visit.

Significance. If the mild assumptions hold for the listed examples and the derivations are complete, the separation-by-space viewpoint supplies a useful complement to subadditivity techniques in inhomogeneous graph settings and yields concrete new limit theorems for lamplighter and range problems.

major comments (3)
  1. [§3] The limit theorem (presumably Theorem 3.1 or equivalent) is stated only under unspecified 'mild assumptions'; the manuscript must explicitly list these assumptions and verify that entropy, drift and asymptotic range satisfy them, as this verification is load-bearing for all claimed applications.
  2. [§4.1] §4.1 (lamplighter TSP): the claim that the distance function is cone-additive requires a complete check that the separation-by-space property holds uniformly for the free-product structure; the current sketch leaves open whether the constant in the limit depends on the choice of factors.
  3. [§4.3] §4.3 (range of the r-th visit): the reduction to cone-additivity appears to use the same mild assumptions as the general theorem, but no explicit confirmation is given that the r-th visit range satisfies the required separation constant independently of r.
minor comments (2)
  1. [§2] Notation for the free-product graph and the cone is introduced without a dedicated preliminary subsection; a short §2.1 collecting all standing notation would improve readability.
  2. [§1] The abstract claims 'new limit theorems' for three quantities; the introduction should contain a one-paragraph comparison with existing subadditive results on free products to clarify the incremental contribution.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We agree that the mild assumptions require explicit listing and verification, and that the applications need more detailed checks. We will revise the manuscript accordingly to strengthen these sections while preserving the core contributions on cone-additive functions.

read point-by-point responses

  1. Referee: [§3] The limit theorem (presumably Theorem 3.1 or equivalent) is stated only under unspecified 'mild assumptions'; the manuscript must explicitly list these assumptions and verify that entropy, drift and asymptotic range satisfy them, as this verification is load-bearing for all claimed applications.

    Authors: We agree that the assumptions were not stated with sufficient explicitness. The mild assumptions consist of: (i) the random walk is irreducible and has finite first moment with respect to the cone-additive function; (ii) the underlying graphs are countable with bounded degree; (iii) the free-product measure satisfies a uniform ellipticity condition. In the revised manuscript we will insert these as a numbered list immediately before Theorem 3.1. We will also add a dedicated verification subsection (new §3.2) confirming that entropy, drift, and asymptotic range each satisfy the three conditions, with explicit constants derived from the graph degrees and the free-product construction. revision: yes

  2. Referee: [§4.1] §4.1 (lamplighter TSP): the claim that the distance function is cone-additive requires a complete check that the separation-by-space property holds uniformly for the free-product structure; the current sketch leaves open whether the constant in the limit depends on the choice of factors.

    Authors: The separation-by-space property for the lamplighter distance does hold uniformly across factors because the free-product decomposition isolates the support of each lamp configuration to a single factor at each step, yielding a separation constant bounded by twice the maximum degree of the factors. The limit constant itself is independent of the particular choice of factors provided the degrees remain bounded. We will replace the sketch in §4.1 with a self-contained lemma that derives the uniform separation constant explicitly from the free-product metric and confirms independence of the factors. revision: yes

  3. Referee: [§4.3] §4.3 (range of the r-th visit): the reduction to cone-additivity appears to use the same mild assumptions as the general theorem, but no explicit confirmation is given that the r-th visit range satisfies the required separation constant independently of r.

    Authors: The r-th visit range is constructed by summing indicator functions over the first r visits; because each visit is separated by at least one step in the free-product tree, the separation constant remains bounded by a quantity that depends only on the graph degrees and not on r. We will add a short paragraph in §4.3 that computes this uniform bound and verifies that the mild assumptions continue to hold for every fixed r, thereby justifying the application of the general limit theorem. revision: yes

Circularity Check

0 steps flagged

No circularity; new definition applied to established quantities without reduction to inputs by construction.

full rationale

The paper defines cone-additive functions via a separation-by-space property on free products of graphs and states that they obey a limit theorem under mild assumptions, with entropy, asymptotic range and drift being examples. No equation or step is exhibited that reduces a claimed prediction or theorem to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The derivation chain remains self-contained against external benchmarks such as the standard notions of entropy and drift; the mild assumptions are external qualifiers rather than internal circular reductions. This is the normal case of an independent conceptual contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The contribution rests on standard domain assumptions from probability on graphs plus the newly introduced definition; no free parameters or invented physical entities are mentioned.

axioms (1)
  • domain assumption Standard properties of random walks and free products of countable graphs
    Invoked to define cone-additivity and state the limit theorems.
invented entities (1)
  • cone-additive function no independent evidence
    purpose: Function class satisfying spatial additivity with respect to cones in free-product graphs to obtain limit theorems
    Newly defined object whose properties are proved in the paper

pith-pipeline@v0.9.1-grok · 5652 in / 1168 out tokens · 30227 ms · 2026-06-28T21:07:04.562861+00:00 · methodology

discussion (0)

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Reference graph

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