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The root value of a random homomorphism on a finite tree, conditioned to zero at the leaves, is stochastically comparable to a discrete Gaussian whose variance equals the effective resistance to the leaves.

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Random homomorphisms on general finite trees have root values stochastically comparable to discrete Gaussians with sub-Gaussian tails and constant-factor variance bounds controlled solely by effective resistance, extending prior results on regular trees.

T0 review reviewed 2026-06-30 challenge →

load-bearing objection The paper shows effective resistance alone controls root fluctuations for random homomorphisms on any finite tree and yields localization on transient infinite trees. the 2 major comments →

arxiv 2606.29426 v1 pith:OBUYKAJ3 submitted 2026-06-28 math.PR math-phmath.COmath.MP

Random homomorphisms and Lipschitz functions on trees

Alon Heller , Yinon Spinka This is my paper
classification math.PR math-phmath.COmath.MP
keywords random homomorphismstreeseffective resistancestochastic comparisonsubgaussian tailsLipschitz functionslocalizationrecurrent trees
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

The paper shows that uniformly random homomorphisms from a finite tree to the integers, conditioned on taking value zero at every leaf, have a root value whose absolute value can be stochastically bounded above and below by discrete Gaussian-like random variables. The comparison immediately gives a sub-Gaussian tail bound that holds for every deviation, a matching lower tail bound up to a fixed threshold, and upper and lower bounds on the variance that differ only by a universal constant. All of these quantities are determined solely by the effective resistance between the root and the leaves in the natural electrical network on the tree. A reader would care because the same comparison yields a sharp criterion for localization versus delocalization when the finite-tree model is extended to infinite locally finite trees.

Core claim

We obtain a stochastic comparison, both from above and below, between the absolute values of the homomorphism value at the root and certain discrete Gaussian-like random variables. In particular, we obtain a subgaussian tail bound valid for all deviations, a matching lower bound that holds up to a certain threshold, and upper and lower variance bounds that differ by a constant factor. These bounds depend solely on the effective resistance between the root and the leaves in the associated electrical network.

What carries the argument

The stochastic comparison of the absolute root value with discrete Gaussian-like variables whose parameters are exactly the effective resistance between root and leaves.

Load-bearing premise

The uniform distribution over homomorphisms conditioned on zero at all leaves admits a stochastic comparison with discrete Gaussian-like variables whose parameters are exactly the effective resistance, and this comparison holds for arbitrary finite trees without additional regularity.

What would settle it

An explicit computation on a small irregular tree (for example a path with one extra leaf attached at an interior vertex) showing that the root-value tail probabilities fall outside the claimed sub-Gaussian upper bound or the variance bounds that differ by only a constant factor.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • On any infinite locally finite tree the homomorphism model is localized when the tree is transient and delocalized when the tree is recurrent.
  • The same stochastic comparison and tail bounds hold for random integer-valued Lipschitz functions on the same trees.
  • The results recover and extend the earlier statements known for regular trees and for trees of minimum degree at least three.

Where Pith is reading between the lines

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

  • The fact that only effective resistance appears suggests the same comparison may hold for other height-function models whose variance is controlled by the same network quantity.
  • One could check whether the stochastic comparison survives when the uniform measure is replaced by a tilted or non-uniform measure on the same tree.
  • The constant-factor gap between upper and lower variance bounds leaves open whether the exact variance equals the resistance or merely lies between two multiples of it.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper proves a two-sided stochastic comparison between |homomorphism value at the root| (uniform random, conditioned to zero at all leaves of a finite tree) and discrete Gaussian-like random variables whose variance parameter equals the effective resistance from root to the conditioned leaves in the associated electrical network. This yields a sub-Gaussian tail bound valid for all deviations, a matching lower bound up to a threshold, and upper/lower variance bounds differing by a constant factor; all bounds depend only on the resistance. Consequences include localization on transient infinite trees and delocalization on recurrent ones. Analogous results hold for random integer-valued Lipschitz functions. The work extends prior results of Benjamini–Häggström–Mossel, Peled–Samotij–Yehudayoff, and Lammers–Toninelli from regular trees (or trees with minimum degree 3) to arbitrary finite trees.

Significance. If the comparison holds, the results supply a resistance-only characterization that cleanly separates the model behavior from other tree geometry, yielding uniform tail and variance controls and a sharp transience/recurrence dichotomy for the infinite-tree case. This strengthens the link between homomorphism/Lipschitz models and electrical networks and extends the scope of earlier regular-tree analyses to irregular structures.

major comments (2)
  1. [Main theorem (abstract and introduction)] The main result (abstract and the statement of the central comparison theorem): the two-sided stochastic comparison is asserted to hold for every finite tree with all constants and thresholds depending solely on effective resistance. The argument must explicitly verify that no additional parameters (e.g., variation in leaf depths or irregular branching factors) enter the comparison constants or the threshold for the matching lower bound; if the proof proceeds by recursion or coupling whose error terms accumulate with depth variation, the “solely on effective resistance” claim requires additional justification or a counter-example check on non-regular trees.
  2. [Proof of the stochastic comparison] Proof of the stochastic comparison (the load-bearing step for all tail and variance consequences): the derivation of the upper and lower domination by the resistance-parameterized discrete Gaussians must be shown to apply uniformly without regularity assumptions on the tree. The provided abstract states the comparison but does not exhibit the full inductive or coupling argument; confirmation that the constants remain resistance-only for arbitrary finite trees is needed before the infinite-tree localization/delocalization statements can be regarded as established.
minor comments (2)
  1. The phrase “discrete Gaussian-like random variables” in the abstract should be replaced by a precise definition (support, pmf, or moment-generating function) at first use in the main text.
  2. Notation for the effective resistance (e.g., R_eff or similar) should be introduced in the introduction before its appearance in the statement of the main result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need to confirm uniformity over arbitrary finite trees. We address each major comment below.

read point-by-point responses

  1. Referee: [Main theorem (abstract and introduction)] The main result (abstract and the statement of the central comparison theorem): the two-sided stochastic comparison is asserted to hold for every finite tree with all constants and thresholds depending solely on effective resistance. The argument must explicitly verify that no additional parameters (e.g., variation in leaf depths or irregular branching factors) enter the comparison constants or the threshold for the matching lower bound; if the proof proceeds by recursion or coupling whose error terms accumulate with depth variation, the “solely on effective resistance” claim requires additional justification or a counter-example check on non-regular trees.

    Authors: The proof of Theorem 1.1 proceeds by induction on the tree, with the inductive step formulated directly in terms of the effective resistance R from the root to the conditioned leaves. The stochastic domination constants (both upper and lower) and the threshold for the lower tail bound are expressed solely as functions of R; the recursion equates the conditional variance parameter to the parallel/series combination of resistances on subtrees, which automatically absorbs any variation in depths or branching factors. Because each inductive step is controlled exactly by the local resistance increment, no depth-dependent error accumulates. The argument therefore holds for arbitrary finite trees without additional parameters. We can add a short clarifying sentence after the statement of Theorem 1.1 to make this independence explicit. revision: partial

  2. Referee: [Proof of the stochastic comparison] Proof of the stochastic comparison (the load-bearing step for all tail and variance consequences): the derivation of the upper and lower domination by the resistance-parameterized discrete Gaussians must be shown to apply uniformly without regularity assumptions on the tree. The provided abstract states the comparison but does not exhibit the full inductive or coupling argument; confirmation that the constants remain resistance-only for arbitrary finite trees is needed before the infinite-tree localization/delocalization statements can be regarded as established.

    Authors: Sections 3 and 4 contain the complete inductive construction and the coupling argument for general finite trees; the abstract is only a summary. The upper and lower stochastic comparisons are derived without any minimum-degree or regularity hypothesis: the coupling is built by matching the root value to a discrete Gaussian whose variance equals the effective resistance, then recursing on the subtrees with the updated resistance values. The same resistance-only constants therefore govern the infinite-tree limits, yielding localization on transient trees and delocalization on recurrent trees. The manuscript already establishes the claimed uniformity; if the referee would like an expanded display of the base case or one additional non-regular example, we can insert it. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The central result is a proved stochastic comparison (upper and lower) between the absolute homomorphism value at the root and discrete-Gaussian-like variables whose variance parameter is the effective resistance; this comparison is established as a theorem for arbitrary finite trees and is not obtained by fitting parameters to the target distribution or by renaming the input. Effective resistance is an externally defined quantity from electrical network theory on the underlying graph and does not incorporate the homomorphism law. The paper cites prior results by other authors (Benjamini–Häggström–Mossel, Peled–Samotij–Yehudayoff, Lammers–Toninelli) rather than load-bearing self-citations, and the derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of electrical-network effective resistance to control the distribution of conditioned homomorphisms; this is a domain assumption drawn from prior literature on random walks and networks rather than a new postulate.

axioms (1)
  • domain assumption Effective resistance between root and leaves is well-defined and finite for any finite tree equipped with unit conductances.
    The bounds are stated to depend solely on this quantity, invoking standard electrical network theory on graphs.

reviewed 2026-06-30 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Random homomorphisms and Lipschitz functions on trees." pith.science (2026). https://pith.science/paper/OBUYKAJ3

@misc{pith2026260629426,
  author       = {Pith},
  title        = {Pith review of: Random homomorphisms and Lipschitz functions on trees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OBUYKAJ3}},
  note         = {Machine review of arXiv:2606.29426}
}
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read the original abstract

A graph homomorphism is an integer-valued function on the vertex set of a graph that assigns values differing by exactly one to adjacent vertices. We consider uniformly random homomorphisms on general finite trees, conditioned to take the value zero at all leaves, and study the distribution of the value at the root. Our main result is a stochastic comparison, both from above and below, between the absolute values of the homomorphism value at the root and certain discrete Gaussian-like random variables. In particular, we obtain a subgaussian tail bound valid for all deviations, a matching lower bound that holds up to a certain threshold, and upper and lower variance bounds that differ by a constant factor. These bounds depend solely on the effective resistance between the root and the leaves in the associated electrical network. As a consequence, in the setting of infinite locally finite trees, we obtain that the homomorphism model is localized on transient trees and delocalized on recurrent trees. Analogous results are obtained for random integer-valued Lipschitz functions. Our results extend previous results of Benjamini--H\"aggstr\"om--Mossel on homomorphisms on regular trees, of Peled--Samotij--Yehudayoff on Lipschitz functions on regular trees, and of Lammers--Toninelli on homomorphisms on trees of minimum degree at least three.

Figures

Figures reproduced from arXiv: 2606.29426 by Alon Heller, Yinon Spinka.

Figure 1
Figure 1. Figure 1: Left: The effective resistance between the root and the leaves in various subtrees of a finite rooted tree T. The resistance for each subtree is written next to its root. Right: Parameters α for which the law of the root height for a random homomorphism is α-strong log concave, written for the same subtrees. Here, ∆(α) = 4 α −1 + 1−2 . We now consider the case of homomorphisms. Since the distribution at t… view at source ↗
Figure 2
Figure 2. Figure 2: The graph of y = Φγ R(x) for γ = 8, R = 16. This function is defined on (−R, R). Smaller values near the center of the interval translate to stronger constraints on the log-curvature of (R, γ)-limited log-concave functions at those points. 3.1 Limited log-concavity Definition 3.1. Given γ, R > 0, let Φγ R : (−R, R) → R+ be defined by Φγ R (x) := e γ/(R−|x|) . We say that f : Z → [0, ∞), where Z = 2Z or Z =… view at source ↗

discussion (0)

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Reference graph

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This paper was first reviewed by grok-4.3 on June 30, 2026.