{"id":"0af1a572-4963-4c9c-b61b-ca5c8f308582","arxiv_id":"2506.02666","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Analytic bounds and asymptotics for a RIS-aided link suffering inter-operator interference from multiple independent RISs under correlated Nakagami-m fading.","lead":"This paper derives closed-form bounds and asymptotic formulas for the spectral efficiency of a wireless link helped by one operator's smart surface but disturbed by other operators' independently controlled surfaces. It covers spatially correlated fading and random interference timing, and it argues such interference becomes negligible for large surface arrays.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The headline asymptotic claim is not supported: Eq. (20) cannot yield the stated '< pM0' linear scaling under the paper's own moment formulas; for perfect CSI and nonzero-mean Nakagami entries, E[X^2]=Tr[R_h R_g] is O(M0^2), not O(M0), and the 'Y→0' step ignores Var[Y]=Σ M_i/2.","rationale":"The reader's weakest_assumption is the unvalidated Gamma approximation in Eqs. (9)-(10). That is a legitimate concern for the outage integral and the lower bound, and I would keep it as a secondary issue. But the single most load-bearing issue is the asymptotic result, because it is stated as an exact limit, it is used for the paper's headline insights (IOI vanishes, linear scaling, spatial correlation has minor impact), and it is internally inconsistent with the paper's own moment equations. Deriving Eq. (20) requires X^2→E[X^2]. Under Eq. (6), for θ_bar=1 that limit is Tr[R_h R_g]. If R_h and R_g are the physical Nakagami amplitude correlation matrices of Eq. (7), Tr[R_h R_g] is quadratic in M0 because off-diagonal entries are the positive product of means; the '<pM0' bound is simply false. If instead R=I is assumed for independence, the same equations imply E[X^2]=M0 < E[X]^2 ≈ 0.616M0^2 for m=1, a variance contradiction. There is no reading in which Eqs. (5), (6), and (20) are simultaneously true. The 'Y→0' step is a further independent error: Eq. (11) gives Y→CN(0, Σ M_i/2), not 0, so the asymptotic removal of IOI is not justified at face value. I therefore see the paper as needing a substantive correction to its asymptotic contributions; the finite-array bounds may survive, which is why I would not move the reader's CONDITIONAL verdict to REJECT. The concrete numerical check above would settle the scaling question definitively.","tokens_in":10335,"tokens_out":16456,"duration_ms":168222,"concrete_test":"Use the paper's Eq. (7) with m=1 and c_{i,j}=0 for i≠j to form R_h=R_g with diagonal 1 and off-diagonal π/4. For M0=100 and θ_bar=1, compute Eq. (20): γ_asym = p·Tr[R_h R_g] = p(100 + 9900(π/4)^2) ≈ 6206p. If the authors instead intend R=I, run a Monte Carlo of X = Σ_{i=1}^{100} a_i b_i with independent Nakagami(m=1) amplitudes: the averaged X^2 will be ≈6206, or at least far above 100, so either way the claimed '<100p' linear scaling is falsified. This same simulation also checks whether Eq. (6) is the correct second moment of X.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing weakness is the asymptotic analysis in Section III, not the Gamma approximation. The paper's headline outcome—IOI vanishes and SNR scales linearly—rests on Eq. (20), but Eq. (20) contradicts the paper's own definitions. For perfect CSI (θ_bar=1), Eq. (6) gives E[X^2]=Tr[R_h R_g]. With the physically consistent independent-Nakagami amplitude model used in Eq. (5), off-diagonal entries of R_h and R_g are E[h_i]E[h_j]=E[h]^2>0 and E[g_i]E[g_j]=E[g]^2>0, so Tr[R_h R_g]=M0+M0(M0−1)E[h]^2E[g]^2=O(M0^2). Eq. (20) is then p[M0(1−θ_bar^2)+O(M0^2)θ_bar^2], which exceeds pM0, so the stated inequality '<pM0' is false. If instead the authors intend R_h=R_g=I for independence, as the text states, then E[X^2]=M0 but Eq. (5) gives E[X]=M0E[h]E[g]θ_bar, so E[X^2] ≥ E[X]^2 is violated for θ_bar=1 and large M0. Either reading is internally inconsistent. Separately, the step 'Y→E[Y]=0' is not valid: Y is zero-mean with variance Σ M_i/2, so by Eq. (11) it converges to CN(0, Σ M_i/2), not to 0; interference can be neglected only if the desired term grows faster, which Eq. (20)'s claimed linear scaling explicitly denies. Thus the central asymptotic claim and the engineering intuition built on it are not established, even though the finite-array bounds may survive.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a point-to-point link assisted by a reference RIS in the presence of N independently operated RISs, which create inter-operator interference (IOI). The desired cascade is modeled over spatially correlated Nakagami-m fading with coherent phase alignment and a von-Mises phase error, while the interfering cascades are modeled via a CLT as zero-mean complex Gaussian. The desired channel amplitude is approximated by a Gamma distribution matched to its first two moments. Closed-form approximations are derived for the outage CDF, for lower and upper bounds on spectral efficiency, and for an asymptotic large-array expression. Monte Carlo simulations are used to validate the analytical curves. The advertised conclusions are that IOI vanishes as RIS arrays grow and that the SNR scales linearly with the number of elements.","tokens_in":10712,"tokens_out":11866,"duration_ms":115634,"significance":"If the moment-based analysis were rigorous, the finite-array part would be a useful extension of the authors' previous work to correlated Nakagami-m channels and asynchronous, multiple IOI sources. The derivations of the first two moments of the desired channel are explicit, no parameters are fitted to simulations, and the closed-form bounds are numerically supported in Figs. 3-6. The central weakness is that the asymptotic section contains an internal inconsistency that invalidates two headline claims, and all closed-form results are proven for a Gamma approximation rather than for the true channel distribution. These issues are load-bearing because the paper's stated engineering insights depend on them.","major_comments":[{"comment":"Eq. (20) contradicts the paper's own moment formulas. For perfect CSI (theta_bar=1), Eq. (6) gives E[X^2]=Tr[R_h R_g]. Using Eq. (7) with the stated Nakagami model, the diagonal entries of R_u are 1 (E[u_i^2]=1) and the off-diagonal entries are E[u_i]E[u_j]=E[u]^2>0, so in the independent case Tr[R_h R_g]=M0+M0(M0-1)E[h]^2 E[g]^2=O(M0^2). Eq. (5) gives E[X]=M0 E[h] E[g], so for large M0 the desired term grows quadratically, and the inequality gamma->pTr<pM0 in Eq. (20) is false. If, alternatively, one takes R_h=R_g=I so that Tr=M0, then E[X^2]=M0 is incompatible with the nonzero mean in Eq. (5) because Var[X] would become negative for large M0. Thus the asymptotic expression, and the linear-scaling/IOI-vanishing interpretation attached to it, is not supported by the model.","section":"Section III, Eq. (20)"},{"comment":"The step 'Y->E[Y]=0' is not valid. Eq. (11) states that Y is approximately CN(0, sum_i M_i/2); as M_i grows, the variance of Y grows, so a zero-mean random variable does not converge to zero in distribution. At best one could argue that Y is negligible relative to a desired term that grows faster, but Eq. (20) explicitly claims only linear growth of the desired term, so the inference that IOI vanishes is unsupported. The variance of Y is instead incorporated correctly in the finite-array expressions (14) and (19), and those parts should be retained if the asymptotic claims are reworked.","section":"Section III, asymptotic paragraph above Eq. (20)"},{"comment":"All closed-form results (outage CDF, bounds, and the Meijer G expression) are derived for the Gamma distribution obtained by matching only E[X] and E[X^2]. No analytical argument is given that this moment-matched Gamma approximates the true distribution of X to sufficient accuracy, nor that the resulting 'bounds' are actual bounds for the true spectral efficiency. The Monte Carlo agreement in Figs. 3-6 supports practical usefulness, but for the advertised claim of tight lower and upper bounds, an error bound or an alternative justification is needed, especially in the small-array (M0=4) and strong-correlation regimes.","section":"Section III, Eqs. (9)-(10) and (13)-(19)"},{"comment":"The definition of X as |h0^T g0| e^{j theta} (a single global phase error) is not consistent with the derivation of E[X^2] in Eq. (6), which expands sums of cos(phi_i) and sin(phi_i) as though there were independent per-element phase errors. With a single global phase, E[X^2] = E[(h0^T g0)^2] E[e^{2j theta}], which depends on E[e^{2j theta}], not on theta_bar^2 as in Eq. (6). The notation should be aligned before the moment expressions can be regarded as self-consistent.","section":"Section III, Eqs. (5)-(6)"}],"minor_comments":[{"comment":"The curve labeled Casy in the figure is never defined in the text; either define it as the asymptotic expression following Eq. (20) or relabel it.","section":"Section IV, Fig. 3"},{"comment":"The substitution 'e^{-x/beta} <- x' is unclear; write t=e^{-x/beta} (or x=beta ln(1/t)) explicitly.","section":"Section III, Eq. (13)"},{"comment":"The symbol Gamma(.) is overloaded (Gamma function versus exponential integral or incomplete Gamma); define the intended special function at first use.","section":"Section III, Eq. (18)"},{"comment":"The phrase 'unit scale' should be stated as Omega=1 with E[u_i^2]=1, since Eq. (7) relies on this normalization.","section":"Section II, channel model"}],"recommendation":"reject","confidential_remarks":"The asymptotic contradiction in Eq. (20) is not a local typo: it is highlighted in the introduction and concluding remarks as the IOI-vanishing and linear-scaling results, and it is contradicted by the paper's own E[X] and R_u definitions. Re-scoping the paper as a purely finite-array approximate analysis without the asymptotic claims would require substantial rewriting and would remove a headline contribution. The unvalidated Gamma approximation compounds this problem, since the closed-form bounds are not proven bounds for the true channel distribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper tackles a real problem—multi-operator RIS interference under spatial correlation and asynchronous timing—and the finite-array part is largely sound. The extension of the authors' earlier independent-fading model to correlated Nakagami-m channels is genuine, the moment derivations in Eqs. (5)-(6) are correct for the stated envelope model, and the Monte Carlo agreement in Figs. 3-6 is encouraging. The Jensen-based bounds and the Meijer G expression are standard machinery, but applied to a new scenario, and the numerical validation gives them credibility. The soft spot is the asymptotic section, and it is not minor. Eq. (20) claims gamma -> p[M0(1-theta_bar^2) + Tr[R_h R_g] theta_bar^2] < pM0, with the interpretation that SNR scales linearly and IOI vanishes. But under the paper's own model, Tr[R_h R_g] contains off-diagonal terms of the form E[h_i h_j] E[g_i g_j]. For independent Nakagami envelopes these equal mu_h^2 mu_g^2 > 0, so Tr[R_h R_g] = M0 + M0(M0-1) mu_h^2 mu_g^2 = O(M0^2). For perfect CSI (theta_bar=1), E[X^2] is O(M0^2), so the claimed linear scaling is false. If one instead forces R_h = R_g = I, then E[X]^2 > E[X^2] for large M0, violating E[X^2] >= E[X]^2. Either reading is internally inconsistent. The step Y -> E[Y] = 0 is also wrong: by the paper's own CLT, Y converges to CN(0, sum M_i/2), not zero. The Gamma approximation that underpins all closed-form bounds is a second concern. It matches two moments, but there is no error control, and the outage integral and bounds hold for the approximate distribution, not the true one. For the tested parameters it works well, but for very small arrays or strong correlation it could drift. This is a known limitation and should be acknowledged. Overall, the paper deserves serious peer review rather than a desk reject. The problem is relevant, the finite-array bounds appear useful, and the errors are specific and fixable. I recommend major revision: correct or remove the asymptotic scaling claim, fix the Y convergence argument, and add a discussion of the Gamma approximation's validity range. The qualitative idea that IOI becomes relatively unimportant for large arrays may survive, but the current support is not sound.","headline":"The finite-array analysis and simulation study are worth a look, but the headline asymptotic claim (linear SNR scaling, IOI vanishing) is contradicted by the paper's own moment formulas and needs fixing before the results are used.","tokens_in":741,"tokens_out":2285,"would_cite":false,"duration_ms":74748,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94A40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes closed-form bounds and an asymptotic law for spectral efficiency in multi-operator RIS systems with uncontrolled inter-operator interference, showing the interference vanishes and SNR grows only linearly with array…","keywords":["reconfigurable intelligent surfaces","inter-operator interference","spatial correlation","Nakagami-m fading","spectral efficiency bounds","channel hardening","asymptotic analysis","multi-operator wireless networks"],"falsifier":"Run a Monte Carlo simulation of the exact model with a small array such as $M_0=4$ and high spatial correlation $\\rho=0.9$, comparing the empirical distribution of $X$ with the Gamma fit of Eqs. (9)-(10) using a Kolmogorov-Smirnov test; a clear mismatch would show the bounds in (14) and (16) do not follow from the true channel statistics.","tokens_in":10078,"feed_emoji":"📡","tokens_out":4402,"duration_ms":39538,"temperature":0.7,"pith_summary":"This paper studies a wireless setting where several operators each deploy their own reconfigurable intelligent surface (RIS), and the rapidly changing phases of the surfaces controlled by other operators create uncontrolled interference for the reference user. The authors try to establish that this inter-operator interference, although random and asynchronous, can still be analyzed with useful closed-form performance expressions. Assuming spatially correlated Nakagami-m fading on the reference links, they derive tight lower and upper bounds on spectral efficiency, an outage probability integral, and an asymptotic formula for large RIS arrays. The asymptotic result shows the interfering terms vanish and the signal-to-noise ratio grows only linearly with the number of RIS elements, with spatial correlation entering only through a matrix trace.","feed_headline":"As RIS arrays grow, inter-operator interference fades away","feed_subtitle":"New closed-form bounds show SNR grows only linearly with array size, not quadratically, under correlated fading.","key_machinery":"The argument rests on two distributional approximations. The desired combined channel $X = |\\mathbf{h}_0^T\\mathbf{g}_0|e^{j\\theta}$ is approximated by a Gamma distribution with shape $\\alpha = E[X]^2/\\mathrm{Var}[X]$ and scale $\\beta = \\mathrm{Var}[X]/E[X]$; the interference term $Y$ is approximated as zero-mean circularly symmetric complex Gaussian with variance $\\sigma^2 = \\sum_i M_i/4$ via the Lyapunov central limit theorem. These feed a conditional noncentral chi-square model for $|X+Y|^2$, yielding an outage integral, Jensen-based spectral-efficiency bounds, and a Meijer-G expression for the lower bound.","core_discovery":"The main discovery is that the large-array behavior of the system is governed by the second moment of the coherently combined reference channel, $E[X^2] = M_0(1-\\bar{\\theta}^2) + \\mathrm{Tr}[\\mathbf{R}_h\\mathbf{R}_g]\\bar{\\theta}^2$, so the received SNR tends to $\\gamma \\to p[M_0(1-\\bar{\\theta}^2) + \\mathrm{Tr}[\\mathbf{R}_h\\mathbf{R}_g]\\bar{\\theta}^2]$. Inter-operator interference disappears in this limit because the uncontrolled phases average out, and the SNR scales linearly with $M_0$ rather than quadratically, the quadratic scaling being impossible when random inter-operator phases are present. Spatial correlation enters only through the trace term, which explains why correlation has a small effect, particularly under near-line-of-sight propagation.","pith_inferences":["If the Gamma approximation for $X$ is replaced by the exact distribution, the same Jensen-based bounding technique would still apply and would likely tighten the lower and upper bounds for small arrays; the paper's numerical tests suggest this direction but do not prove it.","The linear SNR scaling may be a general ceiling for any RIS-assisted link perturbed by uncontrolled phase changes, not only those caused by other operators; testing with a single RIS whose own phase noise is random would separate the multi-operator effect from a pure phase-noise limit.","A direct extension would be to replace the moment-matched Gamma with a numerically evaluated characteristic function to obtain exact outage and spectral-efficiency curves; the paper's asymptotic claim would remain unchanged, but the small-array bounds could be made rigorous.","The beneficial effect of many interference sources suggests a possible design principle: intentionally randomizing phases at secondary RISs could add beneficial scattering for nearby users, though the paper does not optimize such a scheme."],"forward_implications":["In the large-array regime, inter-operator interference vanishes and the received SNR grows linearly with the number of RIS elements, not quadratically as in classical single-operator RIS systems without such interference.","Spatial correlation at the RIS has only a minor effect on spectral efficiency, and the gap between correlated and independent fading becomes marginal when propagation approaches line-of-sight conditions.","Increasing the number of distinct uncontrolled interference sources can improve the reference user's spectral efficiency, because the extra sources enrich the scattered environment and add degrees of freedom.","The derived bounds are valid for correlated Nakagami-m fading and for asynchronous inter-operator interference with random on-off timing; for the opposite timing case the presented results serve as lower bounds.","Imperfect channel estimation, captured by the phase-mismatch concentration parameter $\\kappa$, degrades performance, but the bounds remain tight in the large-array regime even under notable CSI errors."],"supporting_citations":[{"why":"Supplies the spatial correlation matrix entries used in Eq. (7) for correlated Nakagami-m channels.","marker":"[12]"},{"why":"Derives the optimal phase-shift configuration that the reference operator applies to its RIS.","marker":"[13]"},{"why":"Establishes the prior inter-operator interference model that this paper extends to correlated fading and asynchronous sources.","marker":"[9]"},{"why":"Justifies that spatial correlation is always present in practical RIS arrays due to their rectangular structure.","marker":"[11]"},{"why":"Provides the closed-form expectation of the logarithm of a noncentral chi-square random variable used in deriving Eq. (18).","marker":"[16]"},{"why":"Supplies the channel-hardening theorem invoked to show that the interference term averages to zero as array size grows.","marker":"[18]"},{"why":"Establishes the channel-hardening scaling used to claim that $X^2$ concentrates on $E[X^2]$ in the asymptotic regime.","marker":"[19]"}],"fun_headline_variants":["Scale up RIS arrays: interference dies, SNR grows linear","Large RIS arrays: linear SNR, no inter-operator crosstalk","Interference vanishes as RIS arrays grow large","RIS scaling: linear SNR, zero interference in the limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"All closed-form results rest on treating the coherently combined desired channel as a Gamma random variable with only its mean and variance matched, an approximation the paper does not prove; if the true distribution differs materially, the claimed tight bounds are not guaranteed.","fun_headline_variants_meta":{"raw":{"variants":["Scale up RIS arrays: interference dies, SNR grows linear","Large RIS arrays: linear SNR, no inter-operator crosstalk","Interference vanishes as RIS arrays grow large","RIS scaling: linear SNR, zero interference in the limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000189,"raw_usage":{"total_tokens":1279,"prompt_tokens":832,"completion_tokens":447,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":448,"completion_tokens_details":{"reasoning_tokens":380}},"tokens_in":448,"tokens_out":447,"duration_ms":4484,"temperature":1.0,"reasoning_tokens":380,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:21:04.567524+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a Monte Carlo simulation of the exact model with a small array such as $M_0=4$ and high spatial correlation $\\rho=0.9$, comparing the empirical distribution of $X$ with the Gamma fit of Eqs. (9)-(10) using a Kolmogorov-Smirnov test; a clear mismatch would show the bounds in (14) and (16) do not follow from the true channel statistics.","supporting_citations":[{"cited_title":"Performance analysis of RIS-assisted full-duplex communication over correlated Nakagami-m fading channel,","cited_arxiv_id":null,"evidence_quote":"Supplies the spatial correlation matrix entries used in Eq. (7) for correlated Nakagami-m channels."},{"cited_title":"Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,","cited_arxiv_id":null,"evidence_quote":"Derives the optimal phase-shift configuration that the reference operator applies to its RIS."},{"cited_title":"Impact of inter-operator interference via reconfigurable intelligent surfaces,","cited_arxiv_id":null,"evidence_quote":"Establishes the prior inter-operator interference model that this paper extends to correlated fading and asynchronous sources."},{"cited_title":"Rayleigh fading modeling and channel hardening for reconfigurable intelligent surfaces,","cited_arxiv_id":null,"evidence_quote":"Justifies that spatial correlation is always present in practical RIS arrays due to their rectangular structure."},{"cited_title":"Expectations of a noncentral chi-square distribution with application to IID MIMO gaussian fading,","cited_arxiv_id":null,"evidence_quote":"Provides the closed-form expectation of the logarithm of a noncentral chi-square random variable used in deriving Eq. (18)."},{"cited_title":"Couillet and M","cited_arxiv_id":null,"evidence_quote":"Supplies the channel-hardening theorem invoked to show that the interference term averages to zero as array size grows."},{"cited_title":"Exact coverage analysis of intelligent reflecting surfaces with Nakagami- m channels,","cited_arxiv_id":null,"evidence_quote":"Establishes the channel-hardening scaling used to claim that $X^2$ concentrates on $E[X^2]$ in the asymptotic regime."}],"review_version":1}