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arxiv: 2605.15447 · v1 · pith:IJWJVJW3new · submitted 2026-05-14 · ❄️ cond-mat.mes-hall · cond-mat.stat-mech

Multifractal and Ergodic Properties of Conductance Fluctuations under Strong Disorder

Marcos A. A. de Sousa , Heitor R. Publio , Henrique A. de Lima , Adauto J. F. de Souza , Fernando A. Oliveira , Anderson L. R. Barbosa This is my paper

Pith reviewed 2026-05-19 15:00 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.stat-mech
keywords conductance fluctuationsmultifractal analysisergodicityAnderson disorderquantum transportmesoscopic systemstight-binding model
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The pith

Conductance fluctuations transition from non-ergodic to ergodic behavior as disorder strength increases while multifractality persists.

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

This paper examines the stochastic properties of conductance fluctuations in two-dimensional tight-binding models with Anderson disorder. It finds that as disorder strength grows, the fluctuations shift from non-ergodic to ergodic behavior, shown by the conductance correlation function decaying. Multifractal characteristics remain across both regimes, but in the strong disorder case, they no longer change when the time series is shuffled, indicating that the distribution of values dominates over temporal correlations. In contrast, weak disorder shows important long-range correlations. These results hold for different lead geometries and highlight connections between ergodicity, multifractality, and rare events in quantum transport.

Core claim

Using standard multifractal analysis on the fictitious time series of conductance, the work demonstrates a transition from non-ergodic to ergodic behavior with increasing disorder strength, marked by the decay of the conductance correlation function. Multifractality continues in both regimes, yet becomes insensitive to shuffling in the ergodic strong-disorder regime, implying distributional effects prevail over temporal organization, whereas long-range correlations matter in the non-ergodic weak-disorder regime.

What carries the argument

The decay of the conductance correlation function combined with the effect of shuffling the time series on multifractal measures, which separates temporal correlations from distributional effects.

If this is right

  • Multifractality in conductance fluctuations is robust across ergodic and non-ergodic regimes in disordered systems.
  • Distributional effects dominate the multifractal properties in the strong-disorder ergodic regime.
  • Long-range temporal correlations are key to multifractality in the weak-disorder non-ergodic regime.
  • Results remain unchanged under variations in lead geometry.

Where Pith is reading between the lines

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

  • The observed separation of distributional and temporal contributions may guide analysis of fluctuation statistics in other mesoscopic transport settings.
  • Similar shuffling tests could be applied to conductance data from real devices to probe the role of rare events.
  • Persistence of multifractality independent of ergodicity state suggests it could serve as a broader diagnostic for disorder effects.

Load-bearing premise

The decay of the conductance correlation function reliably signals a transition to ergodicity and shuffling the time series cleanly separates temporal correlations from distributional effects without introducing artifacts.

What would settle it

Compute the conductance correlation function across increasing disorder strengths in a tight-binding simulation and check whether its decay reliably marks ergodicity while shuffling leaves multifractal spectra unchanged only in the strong-disorder limit.

Figures

Figures reproduced from arXiv: 2605.15447 by Adauto J. F. de Souza, Anderson L. R. Barbosa, Fernando A. Oliveira, Heitor R. Publio, Henrique A. de Lima, Marcos A. A. de Sousa.

Figure 1
Figure 1. Figure 1: FIG. 1. Mesoscopic two-dimensional sample (in blue) con [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Panel (a) shows the conductance as a function of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Panel (a) presents the generalized Hurst exponents, [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Panel (a) illustrates the conductance as a function [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Panel (a) presents the generalized Hurst exponents, [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

Understanding the stochastic properties of conductance fluctuations in disordered mesoscopic systems is fundamental to quantum transport. In this work, we investigate the multifractal and ergodic properties of the fictitious time series of conductance in two-dimensional tight-binding models under varying Anderson disorder. Using standard multifractal analysis, we show that conductance fluctuations exhibit a transition from non-ergodic to ergodic behavior as the disorder strength increases, as evidenced by the decay of the conductance correlation function. Remarkably, multifractality persists in both regimes; however, it becomes insensitive to shuffling in the strong-disorder (ergodic) regime, suggesting that distributional effects dominate temporal organization. On the contrary, in the weakly disordered (non-ergodic) regime, long-range correlations play a significant role. These findings are robust against changes in lead geometry (asymmetric vs. symmetric). Our results provide new insights into the interplay between ergodicity, multifractality, and rare events in disordered quantum transport.

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

2 major / 2 minor

Summary. The manuscript investigates the multifractal and ergodic properties of conductance fluctuations in two-dimensional tight-binding Anderson models. It reports that conductance fluctuations undergo a transition from non-ergodic to ergodic behavior with increasing disorder strength W, as indicated by the decay of the conductance correlation function. Multifractality persists across both regimes, but becomes insensitive to shuffling in the strong-disorder (ergodic) regime, implying that distributional effects dominate over temporal organization, whereas long-range correlations are significant in the weak-disorder regime. The findings are stated to be robust to changes in lead geometry.

Significance. If the numerical observations and their interpretations hold, the work would contribute to understanding the interplay between ergodicity breaking, multifractality, and rare events in mesoscopic quantum transport. The direct simulation approach on tight-binding models provides concrete data on these properties, and the distinction between temporal and distributional contributions via shuffling is a potentially useful diagnostic. However, the central claims rest on the robustness of the correlation-function decay and shuffling procedure, which require explicit validation to be load-bearing.

major comments (2)
  1. [Results on ergodicity (around the correlation-function analysis)] The central claim that decay of the conductance correlation function C(τ) with increasing W marks a genuine non-ergodic to ergodic transition lacks direct validation. The manuscript should demonstrate equivalence (or lack thereof) between time averages and ensemble averages, for instance by reporting the variance of time averages across disorder realizations and showing convergence to zero only in the strong-W regime. No quantitative threshold for sufficient decay is provided, leaving the transition interpretive rather than rigorously established.
  2. [Multifractal analysis with shuffling] The shuffling procedure is used to argue that multifractality becomes insensitive to temporal correlations in the ergodic regime. However, no control tests are shown demonstrating that the shuffled singularity spectrum f(α) converges to the original spectrum specifically when temporal correlations are absent, rather than due to finite-sample bias in the multifractal analysis. This is load-bearing for the claim that distributional effects dominate in the strong-disorder regime.
minor comments (2)
  1. [Abstract and introduction] The term 'fictitious time series' for the conductance is used without an explicit definition of how the time series is constructed from the model parameters or lead configurations; a short clarification in the methods would improve accessibility.
  2. [Methods] System sizes, exact disorder strengths W, number of disorder realizations, and lead geometries (asymmetric vs. symmetric) should be stated with precise values and ranges in a dedicated methods or simulation-details subsection for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major comments below. We have revised the manuscript to incorporate additional validations as suggested.

read point-by-point responses

  1. Referee: [Results on ergodicity (around the correlation-function analysis)] The central claim that decay of the conductance correlation function C(τ) with increasing W marks a genuine non-ergodic to ergodic transition lacks direct validation. The manuscript should demonstrate equivalence (or lack thereof) between time averages and ensemble averages, for instance by reporting the variance of time averages across disorder realizations and showing convergence to zero only in the strong-W regime. No quantitative threshold for sufficient decay is provided, leaving the transition interpretive rather than rigorously established.

    Authors: We agree that direct validation via time-ensemble equivalence would strengthen the ergodicity claim. In the revised manuscript we have added analysis of the variance of time-averaged conductance computed over long fictitious-time windows for individual realizations; this variance, averaged over the ensemble, decreases toward zero only in the strong-disorder regime. We have also introduced an explicit quantitative threshold (C(τ) < 0.05 for τ larger than a cutoff set by the correlation time) to mark the transition. These additions render the identification of the non-ergodic-to-ergodic crossover more rigorous. revision: yes

  2. Referee: [Multifractal analysis with shuffling] The shuffling procedure is used to argue that multifractality becomes insensitive to temporal correlations in the ergodic regime. However, no control tests are shown demonstrating that the shuffled singularity spectrum f(α) converges to the original spectrum specifically when temporal correlations are absent, rather than due to finite-sample bias in the multifractal analysis. This is load-bearing for the claim that distributional effects dominate in the strong-disorder regime.

    Authors: We thank the referee for this suggestion. To exclude finite-sample bias we have performed control tests on synthetic series: long-range correlated fractional Brownian motion (where shuffling visibly alters f(α)) and uncorrelated white noise (where original and shuffled spectra coincide within error bars). These controls are now included in a new subsection; they confirm that the observed invariance under shuffling in the strong-disorder regime arises from the absence of temporal correlations rather than from analysis artifacts. revision: yes

Circularity Check

0 steps flagged

No circularity: results from direct numerical simulation of tight-binding models

full rationale

The paper reports findings from numerical simulations of 2D tight-binding models with Anderson disorder. The claimed non-ergodic to ergodic transition is identified via decay of the conductance correlation function computed on fictitious time series, with multifractal spectra obtained by standard box-counting or moment methods and shuffling applied as a control. These steps are data-driven computations rather than algebraic derivations; no equation reduces to a fitted parameter renamed as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in. The interpretation of correlation decay as an ergodicity marker is an external physical claim, not a definitional tautology internal to the paper's equations. The manuscript is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; no explicit free parameters, invented entities, or non-standard axioms are stated in the summary.

axioms (1)
  • domain assumption Standard multifractal analysis methods can be applied to fictitious conductance time series generated from tight-binding simulations
    Invoked to characterize fluctuations and detect the reported transition and shuffling insensitivity

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