{"id":"eb633d5a-7447-4c04-ac52-3cc14336bd9f","arxiv_id":"1908.06432","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Holographic subregion complexity in a general Vaidya geometry grows linearly at early and intermediate times, then decreases linearly at late time for continuous transitions, with growth rates below the Lloyd bound in most dimensions.","lead":"This paper analytically derives the evolution of holographic subregion complexity during a quench in a general Vaidya-AdS spacetime, identifying three stages: early linear growth, intermediate linear growth with a smaller rate, and late-time linear decrease or growth. It shows that subregion complexity differs from full spacetime complexity even in the large-size limit, which matters for interpreting quantum complexity in strongly coupled field theories.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sec. 2.5 evaluates the volume on the HRT profile without verifying that the surface extremizes the volume functional; the stationarity condition for Γ_A differs from the HRT area equations, so Eqs. (55), (83), and (116) may be off-shell.","rationale":"The paper's central claim is an analytic formula for the evolution of holographic subregion complexity. I read the derivation as a chain: fix the HRT surface, then compute the volume of the codimension-one surface Γ_A bounded by the strip and the HRT surface. For the final quantity to be the subregion CV complexity, Γ_A must be an extremum of the volume functional, not merely a surface that contains the HRT surface as a boundary component. The manuscript does not derive or verify the corresponding stationarity condition. This is more fundamental than the large-size expansion issue: even if the expansion in Eq. (61) were fully controlled, the volume being expanded might still not be the extremal volume. I am not asserting the final rates are false; I am asserting the paper has not established that its computed volume is the holographic complexity. The proposed test is an analytic check that can be performed before any numerical evolution and would settle whether the identification is valid. If the check passes, the conditional acceptance can proceed; if it fails, the central claim is unsupported by the present derivation.","tokens_in":17272,"tokens_out":37526,"duration_ms":376988,"concrete_test":"Compute the first variation of the full volume functional S[V] = ∫ d^{d-1}ξ sqrt{|det g|} for Γ_A at the ansatz V(z,x) = v_HRT(z) in the SAdS-Vaidya metric (2), using the HRT solutions (24)-(27). Evaluate the Euler-Lagrange residual E = ∂_z(∂L/∂V_z) + ∂_x(∂L/∂V_x) - ∂L/∂V at a generic point in the black-hole region, with V(0,x) = t and V(z,x̃(z)) = v_HRT(z). If E ≠ 0 for some compactly supported variation, the surface used in Sec. 2.5 is not extremal and the central volume computation is not the subregion CV complexity.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"Section 2.5 identifies the on-shell volume with the integral of Eq. (38) evaluated on the HRT profile v_HRT(z) from Eqs. (16), (18), and (26)-(27). For the subregion CV conjecture, Γ_A must be an extremum of the volume functional subject to ∂Γ_A = A ∪ γ_A. That stationarity condition is not the HRT area equation (12)-(13). With the general embedding v = V(z,x), the volume integrand also depends on V_x through the cross term g_zx, and the Euler-Lagrange equation contains ∂_x(∂L/∂V_x) as well as a weighted ∂_z term. The paper never writes or solves this equation. The ansatz V_x = 0, V = v_HRT(z), x ∈ [0, x̃(z)] is one particular surface satisfying the boundary conditions, but nothing in the manuscript shows that it is extremal. If it is not extremal, the claimed rates in Eqs. (55), (83), and (116) are the volume of an arbitrary surface, not the holographic subregion complexity. The d = 2 expansion issue raised by the reader is a secondary concern compared with this identification.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies holographic subregion complexity (HSC) under the subregion CV conjecture in a general Vaidya-AdS spacetime describing a thin-shell quench. For a strip subregion of large half-width l, it claims that the evolution of HSC has three distinct stages: an early-time linear growth with rate (1/2)A_A R_AdS^d ω t, an intermediate-time linear growth with a dimension-dependent rate (e.g., Eq. (83) for d≥3), and a late-time stage that is either continued linear growth (discontinuous transition) or linear decrease (continuous transition). The paper also compares the growth rates with the Lloyd bound and argues that subregion CV differs from global CV even in the large-size limit.","tokens_in":17518,"tokens_out":12151,"duration_ms":128463,"significance":"If the central claim were established, the paper would provide a useful analytic characterization of mixed-state complexity growth during holographic thermalization, with explicit dimension-dependent rates that could be checked by numerics and field-theoretic models. The general Vaidya setup, the clear three-stage picture, and the step-by-step asymptotic expansions are valuable. However, the central quantity is computed without solving the volume-extremization problem for Γ_A, and the asymptotic control in the intermediate stage is incomplete. These issues currently block the main claim, so the paper needs substantial revision before its results can be accepted as holographic subregion complexity.","major_comments":[{"comment":"The volume functional is evaluated on the HRT profile v(z)=v_HRT(z) without solving the variational problem for the subregion CV surface. Equation (1) requires Γ_A to extremize the volume subject to ∂Γ_A = A ∪ γ_A. For a general embedding v=V(z,x), the induced metric contains V_x-dependent terms, and the stationarity condition for (38) differs from the HRT area equations (12)-(13). The ansatz V_x=0, V=v_HRT(z) satisfies the boundary conditions but is not shown to be a solution of the volume Euler-Lagrange equation. Unless this gap is closed, the rates in Eqs. (55), (83), and (116) are volumes of an arbitrary off-shell surface, not necessarily the HSC defined by the subregion CV conjecture.","section":"Sec. 2.5, Eqs. (35)-(38)"},{"comment":"The intermediate-stage derivation relies on the hierarchy in Eq. (61) and on keeping only the leading log ϵ terms in t and l. The paper does not bound the subleading corrections. In d=2 this is not a technicality: Eq. (90)-(91) show that the leading linear growth rate vanishes as l→∞, so the claimed linear growth near the critical configuration must be carried by terms of higher order, which are not computed. The text acknowledges this point for d=2, but the abstract and Sec. 3.2 state the three-stage linear-growth picture without qualification for the general Vaidya setup (d≥2). The d=2 case must either be excluded from the claim or treated to the order that supports the linear rate.","section":"Sec. 3.2, Eqs. (61)-(69) and (90)-(91)"},{"comment":"The late-time linear decrease for the continuous transition follows from the sign of C_s. The sign is decided by the inequality I'(z_s)/F'(z_s) > (1/z_s^{d-1}) ∫_0^1 dz z^{-d}/h(z_s z) in Eq. (119), but I'(z_s) and F'(z_s) are defined as limits of expressions that each contain a 1/θ divergence, and the finite parts are not computed. The inequality between the finite parts is not demonstrated; it appears to be inferred from the integrands rather than derived. Since the negative sign of C_s is the basis of the late-time decrease, this step needs a controlled derivation or a numerical check.","section":"Sec. 3.3.2, Eqs. (117)-(119)"},{"comment":"The early-time saturation of the Lloyd bound is enforced by the choice R_AdS/R = (d-1)/(4π²) in Eq. (59). Since R is the free length scale in the CV formula (1), this choice sets the rate to 2M/π by construction. The statement that the Lloyd bound is 'always saturated' at early time is therefore a choice of convention rather than a dynamical result, and the comparison with the Lloyd bound does not provide independent evidence for the proposed growth law.","section":"Sec. 3.1, Eq. (59)"}],"minor_comments":[{"comment":"There is a typo: 'anothor' should be 'another'.","section":"Sec. 2.3, Eq. (15)"},{"comment":"Equation (91) appears garbled in the typeset version: the square-root symbol is rendered as an integral sign. Please correct the typesetting.","section":"Sec. 3.2, Eq. (91)"},{"comment":"The closely related work by Auzzi et al., 'On volume subregion complexity in Vaidya spacetime' (reference [14]), should be discussed explicitly so that the reader can see the relation and the differences between the two computations.","section":"General"},{"comment":"The notation for the two length scales R and R_AdS is confusing: Eq. (1) uses R, while Eq. (2) and most of the paper use R_AdS. Please clarify the distinction and the role of R consistently.","section":"Sec. 1 and Sec. 3.1"}],"recommendation":"major_revision","confidential_remarks":"The main variational gap in Sec. 2.5 is the key issue: if the authors can prove that the HRT profile also extremizes the volume functional, the paper could become acceptable after revision. The d=2 intermediate-stage problem and the sign of C_s also need attention. The Lloyd-bound saturation is a free-parameter artifact and should be presented as such."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou should know two things about this paper: it gives the first analytic three-stage description of subregion complexity growth in general Vaidya backgrounds, and it never checks that the surface whose volume it computes is an extremum of the volume functional. The first is genuine progress; the second is a load-bearing gap.\n\nWhat's new and good: the paper extends the Liu-Suh strategy from entanglement entropy to subregion complexity, computes the early-time rate cleanly (Eq. 55), and derives the intermediate and late-time behavior for large strips, including the discontinuity classification and the explicit difference from global CV. The large-size limit was exactly where the earlier numerics lost control, so this fills a real hole. The derivations are detailed and mostly self-consistent for d ≥ 3.\n\nWhere the soft spots are. The main one is extremality. In Sec. 2.5 the volume is evaluated on the HRT profile v_HRT(z) with V_x = 0. But the stationarity condition for the volume functional contains V_x and differs from the HRT area equations (12)-(13); the paper never writes or solves that Euler-Lagrange equation. The ansatz satisfies the boundary conditions, but that's not enough. If the surface is off-shell, Eqs. (55), (83), and (116) describe the volume of a particular surface, not holographic subregion complexity. This is the stress-test point and, on reading the paper, it holds up.\n\nSecondary: the intermediate/late expansions rely on the asymptotic hierarchy in Eq. (61). That's plausible for d ≥ 3, but in d=2 the leading linear rate vanishes as l→∞ (Eq. 91), so the claimed linear growth near the critical configuration has to be carried by subleading terms that aren't controlled. Also minor: Lloyd-bound saturation at early time follows from choosing R_AdS/R = (d-1)/(4π^2) from their earlier paper, so it's an input rather than a prediction.\n\nOverall: the paper is careful and honest within its chosen framework, but the framework's link to the CV conjecture isn't demonstrated. If the authors explicitly reframe the result as the volume of the region bounded by A and γ_A, the computations stand as useful analytics. If they want to claim it as subregion complexity, they need to face the extremal volume problem.\n\nRecommendation: send to peer review. The gap is real but addressable, and the topic is worth referee time. A serious referee should ask for either a numerical solution of the volume EL equation or a clear redefinition of the quantity.","headline":"Genuinely useful analytic three-stage computation of a subregion volume in Vaidya, but the paper never shows the surface is an extremum of the volume functional.","tokens_in":18060,"tokens_out":12781,"would_cite":false,"duration_ms":119889,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Holographic subregion complexity after a sudden quench grows in three stages: linear growth, slower linear growth, then either linear growth or linear decrease depending on the transition.","keywords":["holographic subregion complexity","subregion CV conjecture","Vaidya-AdS spacetime","holographic thermal quench","HRT surface","Lloyd bound","complexity growth"],"falsifier":"Compute the full $\\Delta V(t)$ numerically in $d=2$ Schwarzschild-AdS for a very large strip half-width $l$ near the critical configuration $z_c=z_c^*(1-\\epsilon)$ and check whether the slope tends to zero as $l\\to\\infty$; a nonzero surviving linear slope would invalidate the claim that the leading rate vanishes, while a zero slope would confirm that the linear growth is carried by uncontrolled subleading terms.","tokens_in":1945,"feed_emoji":"⏱️","tokens_out":6326,"duration_ms":101400,"temperature":0.7,"pith_summary":"The paper tries to establish that, in a general Vaidya-AdS spacetime dual to a sudden quench, the holographic subregion complexity of a large strip-shaped boundary region evolves through three distinct stages. Under the subregion CV (complexity-equals-volume) conjecture, the early stage shows linear growth whose rate is fixed by the asymptotic metric coefficient; an intermediate stage near critical extremal surfaces shows a slower linear growth; and the late stage is either continued linear growth or linear decrease depending on whether the transition is discontinuous or continuous. The authors also compare the growth rates with the Lloyd bound: with a particular choice of length scale, the early-time rate saturates the bound while the intermediate rate always lies below it. Finally, they argue that subregion CV complexity remains different from global CV complexity even when the strip covers the whole boundary, because the tip of the extremal surface still settles to a finite equilibrium value.","feed_headline":"Three linear stages govern complexity growth in a holographic quench","feed_subtitle":"Analytic Vaidya-AdS calculation fixes the growth rates and shows subregion CV never matches global CV even for an infinite strip.","key_machinery":"The central object is the codimension-one extremal volume $\\Gamma_A$ whose boundary is the strip $A$ on the AdS boundary together with the HRT surface $\\gamma_A$; the subregion CV conjecture sets $C_A = V(\\Gamma_A)/(R G_N)$. The argument is carried by the conserved quantity $z^{d-1} L_S = C$, the matching conditions at the null shell $v'_+=v'_-$ and $z'_+ = (1-\\tfrac12 g(z_c))z'_+$, and the critical HRT surface at $z_c=z_c^*(1-\\epsilon)$ near which the function $H(z)$ develops a double zero at $z_m$. This double zero produces a logarithmic divergence in $\\epsilon$ that links time, strip width, and the tip depth $z_t$; expanding time and on-shell volume in $1/z_t$ and $\\epsilon$ around that critical configuration yields the three linear rates.","core_discovery":"On its own terms, the central claim is that in the large-size limit the change in on-shell volume $\\Delta V(t)$ of the extremal surface $\\Gamma_A$ bounded by the strip and its HRT surface follows three analytic regimes. At early times, $\\Delta V(t) = A_A R_{\\mathrm{AdS}}^d (\\omega/2) t + \\cdots$, independent of the strip size, with $\\omega$ fixed by $g(z)\\sim\\omega z^d$ near the boundary. At the intermediate stage, when the HRT surface is near its critical configuration $z_c=z_c^*(1-\\epsilon)$ with $l\\gg t\\gg z_h$, the growth is again linear; for Schwarzschild-AdS in $d\\ge 3$ it reads $\\Delta V(t)=A_A R_{\\mathrm{AdS}}^d \\sqrt{d(d-2)/(2(d-1))}\\, z_h^{-d}\\, t$. At late times, if the transition is discontinuous the growth remains linear all the way to equilibrium, while if it is continuous the subtracted volume decreases linearly with coefficient $C_s$ in Eq. (116), which the paper argues is negative. The reason subregion CV differs from global CV even for an infinite strip is that in the global case the maximal-volume surface is always a Cauchy surface, whereas the subregion surface's tip always descends to a finite equilibrium value $z_s$.","pith_inferences":["The paper does not derive the subregion CA (complexity-equals-action) analogue, but the same three-stage structure should appear there without the arbitrary length-scale ambiguity, so a direct CA calculation would be a natural test of the mechanism.","The late-time linear decrease, if taken literally, conflicts with any monotonic second law of complexity for mixed states; this suggests the apparent decrease may encode reference-state dependence or a limitation of the subregion CV measure rather than a true decrease of pure-state complexity.","In the $d=2$ Schwarzschild-AdS case the leading intermediate growth rate vanishes as $l\\to\\infty$, so the claimed linear behavior must be carried by subleading corrections; exact numerics for large $l$ could show whether a residual linear slope survives or the growth is genuinely sublinear.","Because the early-time rate depends only on $\\omega$ while the intermediate rate depends on $z_h$ and the dimension, comparing thermal and electromagnetic quenches in the same spacetime should give different intermediate rates, providing a sharp observable signature of the quench type."],"forward_implications":["For any thin-shell holographic quench whose metric satisfies the stated properties, a large strip's subregion complexity grows linearly at early and intermediate times, with rates fixed only by the asymptotic falloff $\\omega$ and the final horizon scale $z_h$.","In Schwarzschild-AdS with $d\\ge3$, the early-time rate can saturate the Lloyd bound for a suitable length scale, while the intermediate rate is strictly below the bound for every finite dimension.","Subregion CV complexity is not a large-region limit of global CV complexity: even as the strip covers the whole boundary, the difference persists because the extremal surface's tip behaves differently.","Late-time behavior is controlled by the nature of the transition: a discontinuous jump of the tip gives continued linear growth, while a continuous approach to equilibrium gives a linear decrease near $t_s$.","The equilibrium time of the HRT surface from the referenced analysis applies directly to subregion complexity evolution, since the extremal volume and the surface share the same equilibrium time."],"supporting_citations":[{"why":"Introduces the subregion CV conjecture and the volume functional $C_A=V(\\Gamma_A)/(RG_N)$ that the paper evaluates.","marker":"[3]"},{"why":"Formulates subregion complexity as a measure of the difference between two mixed states, grounding the physical interpretation used here.","marker":"[16]"},{"why":"Previous numerical study of holographic subregion complexity under a thermal quench, whose equilibrium-time result and late-time behavior this paper extends analytically.","marker":"[29]"},{"why":"Supplies the analytic strategy of expanding time and on-shell volume in $1/z_t$ and $\\epsilon$ near the critical HRT surface, plus the equilibrium-time formula.","marker":"[31]"},{"why":"Computes global CV complexity in Vaidya spacetimes and is used for consistency of the early-time rate and the Lloyd-bound comparison.","marker":"[34]"},{"why":"Provides the length-scale choice $R_{\\mathrm{AdS}}/R=(d-1)/(4\\pi^2)$ that makes the early-time rate saturate the Lloyd bound.","marker":"[35]"}],"fun_headline_variants":["Three linear stages in subregion complexity during Vaidya quench","Subregion complexity shows three linear regimes, Lloyd bound only early","Infinite strip subregion CV differs from global CV in Vaidya","Vaidya quench: subregion complexity splits from global CV","Three-phase complexity: Lloyd bound only early, late decline"],"cache_read_input_tokens":20224,"weakest_assumption_plain":"The intermediate- and late-time calculation assumes the hierarchy $z_c^*/z_t$, $z_m/z_t$, and $z_c^*|\\log\\epsilon|$ are all much smaller than one and that the leading logarithmic terms dominate the expansions of $t$ and $l$; in the $d=2$ SAdS case the leading rate vanishes as $l\\to\\infty$, so the claimed linear behavior would have to come from subleading terms that the paper does not systematically control.","fun_headline_variants_meta":{"raw":{"variants":["Three linear stages in subregion complexity during Vaidya quench","Subregion complexity shows three linear regimes, Lloyd bound only early","Infinite strip subregion CV differs from global CV in Vaidya","Vaidya quench: subregion complexity splits from global CV","Three-phase complexity: Lloyd bound only early, late decline"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001098,"raw_usage":{"total_tokens":4623,"prompt_tokens":1026,"completion_tokens":3597,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":642,"completion_tokens_details":{"reasoning_tokens":3508}},"tokens_in":642,"tokens_out":3597,"duration_ms":23339,"temperature":1.0,"reasoning_tokens":3508,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:45:13.998054+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the full $\\Delta V(t)$ numerically in $d=2$ Schwarzschild-AdS for a very large strip half-width $l$ near the critical configuration $z_c=z_c^*(1-\\epsilon)$ and check whether the slope tends to zero as $l\\to\\infty$; a nonzero surviving linear slope would invalidate the claim that the leading rate vanishes, while a zero slope would confirm that the linear growth is carried by uncontrolled subleading terms.","supporting_citations":[],"review_version":1}