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Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in Landau-Ginsburg theory

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Pith's one-line read The paper derives an exact critical measurement rate for an entanglement transition in random tree tensor networks and conjectures the same rate applies to all-to-all monitored circuits.

desk verdict The tree-level result is a genuine exact calculation and the paper deserves a serious referee, but the abstract overstates the circuit-level claim, which rests on an openly conjectural tree-to-circuit transfer that the numerics do not yet confirm. read the letter →

arxiv 2009.11311 v2 pith:PISOTJAF submitted 2020-09-23 cond-mat.stat-mech cond-mat.dis-nncond-mat.str-elhep-thquant-ph

Adam Nahum , Sthitadhi Roy , Brian Skinner , Jonathan Ruhman This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.dis-nncond-mat.str-elhep-thquant-ph MSC 82B2660K3581P40 PACS 05.70.Fh03.67.-a64.60.ah
keywords measurement-inducedphasetransitionentanglementrandomtensornetworkstreeall-to-allquantumcircuitsdirectedpolymeronareplicafieldtheoryforcedmeasurement
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper seeks exactly solvable limits for measurement-induced phase transitions, where monitored quantum circuits are known to have an entangling and a disentangling dynamical phase. Its central result is an exact entanglement transition in random tree tensor networks with bond dimension 2 and U(2)-invariant node tensors: the critical measurement rate for Haar-random unitaries is $r_c = \frac{212 + 75\pi}{362 + 75\pi}$, with the order parameter vanishing as $Z_{\rm typ} \sim \exp(-C/\sqrt{r_c - r})$ on the entangled side. The authors conjecture that this same $r_c$ locates the forced-measurement phase transition in all-to-all Haar circuits, which would be the first exact critical point for a generic, non-Clifford monitored system. The paper also proposes Landau-Ginzburg field theories for the measurement transition and for the forced-measurement/random-tensor-network transition, finding that the two require different replica limits and different field content, and analyzes a classical minimal-cut problem whose percolation exponents apply in high spatial dimensions.

What carries the argument

The load-bearing object is the recursive distribution of the squared minimal singular value $Z_k$ of a depth-$k$ tree tensor network, whose node tensors are built from a Haar-random two-site unitary and, with probability $p$, a projection on one leg. The linearized recursion $Z_{k+1} = \sum_{i=1}^3 A_i Z_k^{(i)}$ (or $0$ with probability $p$) is analyzed through the generating function $G_k(x) = \langle \exp(-e^{-x} Z_k) \rangle$, which becomes a traveling-wave equation in the fictitious coordinate $x$. The two identities $\langle A_i \rangle = 1$ and $\langle A_i^{1/2} \ln A_i \rangle = 0$ imply that the selected wave has decay exponent $\lambda_* = 1/2$ at the transition, giving $p_c = 1 - 1/\sum_i \langle A_i^{1/2} \rangle$. A continuum Fisher-KPP-like reaction-diffusion equation with a position-dependent diffusion coefficient $D(x) = 1 + e^x$ then yields the stretched-exponential critical scaling. The same replica symmetry group $G_N = (S_N \times S_N) \rtimes \mathbb{Z}_2$ is used to write the two candidate Landau-Ginzburg Lagrangians.

What would settle it

Simulate a forced-measurement all-to-all Haar circuit beyond the paper's sizes (for example with stabilizer-based numerics at $N \gtrsim 30$), extract the rate at which the exponential-timescale coefficient $a(r)$ extrapolates to zero, and compare it with $r_c = (212 + 75\pi)/(362 + 75\pi)$. A critical value that differs from $r_c$ beyond finite-size corrections, or a power-law rather than stretched-exponential vanishing of the plateau value $s(r)$, would falsify the tree-to-circuit conjecture.

Watch

Extended reading notes

Core claim

The central discovery is that an entanglement phase transition in a class of random tree tensor networks can be located exactly. The order parameter is the typical value of $Z_k$, the square of the smaller singular value in the Schmidt decomposition between the apex and the base of a depth-$k$ tree. A recursion relation for $Z_k$, linearized near the transition, is mapped to a directed polymer on a tree with random potentials, and two exact identities following from the U(2) invariance of the node tensors---$\langle A_i \rangle = 1$ and $\langle A_i^{1/2} \ln A_i \rangle = 0$---force the selected traveling-wave solution to have decay exponent $\lambda_* = 1/2$ at the critical point. Averaging the constants $A_i$ over Haar-random unitaries yields the closed-form critical rate $r_c = (212 + 75\pi)/(362 + 75\pi) \simeq 0.7490$, below the classical percolation threshold at $0.8$. Using a simplified nonlinear recursion, the paper derives the stretched-exponential scaling $Z_{\rm typ} \sim \exp(-C/\sqrt{r_c - r})$ and conjectures, via a heuristic bound, that the same $r_c$ controls the forced-measurement phase transition in all-to-all Haar circuits, so that the transmitted information per spin and the exponential information timescale vanish very rapidly as $r \to r_c$.

Load-bearing premise

The load-bearing premise is that the forced-measurement all-to-all circuit is tree-like enough for the exact tree critical point, $r_c = (212 + 75\pi)/(362 + 75\pi)$, to be the circuit's transition point; the paper labels this a 'plausible conjecture' and its supporting bound 'far from being a proof'.

Editorial extensions

If this is right

  • The exact tree result gives a parameter-free prediction for the forced-measurement transition in all-to-all Haar circuits: $r_c \approx 0.749$, separating a phase with exponential-in-$N$ information survival from a phase with rapid loss of the initial state.
  • Because $Z_{\rm typ}$ and hence the plateau entanglement density vanish as $\exp(-C/\sqrt{r_c - r})$, the transition is extremely hard to locate by finite-size numerics, which explains the apparent shift of the measured critical rate to smaller values.
  • For tree tensor network states on a chain, the coefficient $c(r)$ of the logarithmic entanglement $S_\ell \sim c(r) \ln \ell$ vanishes exponentially as $r \to r_c$, and the state has $O(1)$ entanglement at the critical point, not super-area-law.
  • In the classical minimal-cut problem, the all-to-all scaling variables are $t/N^{1/5}$ and $(r-r_c)N^{2/5}$, with the minimal-cut tension going as $(r_c - r)^{5/2}$; these exponents also apply to the percolation problem in spatial dimension $d \ge 5$ (with logs at $d=5$).
  • The proposed field theories differ between the MPT (replica limit $N \to 1$, traceless field $X$, upper critical dimension $6$) and the FMPT/RTN (replica limit $N \to 0$, unconstrained field $Y$, upper critical dimension $10$), implying distinct universality classes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the tree-to-circuit conjecture survives more precise tests, it suggests that loop corrections in the all-to-all FMPT are irrelevant or marginal, so the universal critical physics is captured by the tree fixed point rather than by the high-dimensional limit of the Landau theory; a sharper test would be to compute the leading loop correction to the tree recursion in a finite-size circuit.
  • The stretched-exponential vanishing of the order parameter means that any finite-time simulation will see an apparent transition point biased below the true $r_c$, so fitting $a(r)$ and $s(r)$ to $\exp(-C/\sqrt{r_c - r})$ offers a practical extrapolation method for determining $r_c$ from small-system data.
  • The structural difference between the MPT and FMPT field theories suggests that the purification dynamics of a mixed initial state (Born-rule measurements) and the postselected dynamics (forced measurements) may exhibit different critical exponents even in the same all-to-all hardware, a distinction that could be probed in experiments with active feedback or postselection.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript develops several complementary approaches to measurement-induced phase transitions (MPT) and forced-measurement phase transitions (FMPT) in all-to-all random quantum circuits. It first solves a classical min-cut/percolation toy model, obtaining the scaling variables t/N^{1/5} and δr N^{2/5} and the exponent 5/2 for the min-cut tension. It then studies random tree tensor networks with bond dimension 2 and U(2)-invariant node tensors, deriving a linear recursion for the smallest Schmidt weight Z_k. Using a traveling-wave analysis of this recursion, the authors obtain an exact critical point for the tree, r_c = (212+75π)/(362+75π) for Haar unitaries, together with a nonlinear toy-model prediction of exponential critical scaling Z ~ exp(-C/√(r_c-r)). They conjecture that this tree transition controls the FMPT in all-to-all Haar circuits, supported by a one-sided bound and by small-size numerics. The final part constructs two Landau-Ginsburg Lagrangians, L_X for the MPT and L_Y for the FMPT/RTN, with replica limits N→1 and N→0, respectively. The paper is explicit at several points that the circuit identification and the field theories are conjectural.

Significance. The tree tensor network result is significant and, if correct, is one of the few exact entanglement transitions in a finite-bond-dimension random tensor network, with a parameter-free, closed-form threshold for the Haar ensemble. The identities ⟨A_i⟩=1 and ⟨A_i^{1/2} ln A_i⟩=0 give a clean mechanical reason why λ*=1/2 at the transition, and the exact averages in App. C are a strong part of the paper. The classical min-cut/percolation scaling is concrete and testable. The proposed field theories, while explicitly speculative, are thought-provoking and may stimulate further work. The main caveat is that the advertised all-to-all circuit result is not proven: the exact tree result, not the circuit result, is the rigorously established core. No machine-checked proofs or reproducible code are provided, so the analytic derivations and the numerical recursions are the primary evidence.

major comments (3)
  1. [II D, IV K, V B] Equation (12) is rigorously a property of the tree recursion, not of the all-to-all circuit. The transfer is stated in Sec. II D as a 'plausible conjecture', and in Sec. IV K the connecting argument is 'far from being a proof'. The bound in Eq. (113) is one-sided: it gives s(r) ≤ c1 exp(-c2/√(r_c-r)), and the authors themselves note that the analogous classical bound yields only s_cl ≲ (r_c^cl - r)^2 rather than the true exponent 5/2. The circuit numerics in Sec. V B do not discriminate: Fig. 24 shows a(r) vanishing at r appreciably below 0.749, which the authors attribute to exponential flatness but which is equally consistent with a circuit r_c smaller than the tree value. Given that the abstract and introduction claim exact results for all-to-all circuits, the manuscript should either provide a proof of the tree-to-circuit correspondence or clearly demote the circuit identification to a conjecture and revise the abstract accordingly.
  2. [IV H 2, IV H 3, Fig. 14] The exponential scaling Z^typ ~ exp(-C/√(r_c-r)) is obtained from the ad hoc nonlinear recursion Eq. (84), not from the original tree recursion Eqs. (59)-(60). The conjecture that the toy model captures the universal scaling of the actual tree is plausible but unproved, and the numerical support in Fig. 14 is weak in a specific sense: the fitted prefactors D=2.01 (Haar) and D=3.24 (Δt=0.3) differ substantially from the analytic prediction D=1.482 quoted in Eq. (104), with the discrepancy attributed to finite-size effects. This scaling is nevertheless used to obtain Eq. (113) and hence the circuit-level bound. Please either supply a more direct derivation of the exponential form or label it as conjectural at every point where it enters later claims.
  3. [VI E, VI G, VI H] The Landau-Ginsburg Lagrangians L_X and L_Y are presented as candidates, and the authors appropriately caution that they are conjectures. However, a specific tension deserves more discussion: as stated in Sec. VI H, the exact tree transition is not captured by the high-dimensional limit of L_Y, since the tree shows exponential rather than power-law scaling of the order parameter. Since the tree result is one of the paper's main exact results, the field-theory section would be strengthened by a quantitative test of L_X or L_Y against known numerical data or against the tree/continuum exponents, or by an explicit statement of the regime in which each Lagrangian is supposed to apply. Without such a test, the claimed 'surprising difference' between the MPT and the FMPT/RTN field theories remains speculative.
minor comments (5)
  1. [IV F, Eq. (81)] The text states that p_c = (212+75π)/(512+75π) is 'equivalent' to r_c = (212+75π)/(362+75π); this is correct only through the relation p = r/(2-r), and the distinction between p_c and r_c should be stated explicitly to avoid confusion.
  2. [III B] There is a typo in 'the corresponding scailings for operators'; 'scailings' should be 'scalings'.
  3. [IV H 1] The word 'afflicted' in 'The numerical method we use is afflicted by severe finite size effects' is a typographical artifact; it should read 'afflicted'.
  4. [IV H 3, Eq. (89)] The definition G_k(x) = 1 - (p_2 - p_0)/p_2 H_k(x) is ambiguous as printed; adding parentheses around the fraction would improve readability.
  5. [Abstract and II D] The abstract's phrase 'exact results' for all-to-all circuits conflicts with the qualified statement in Sec. II D that Eq. (12) is transferred to the circuit only under a 'plausible conjecture'; the wording should be aligned with the actual status.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the tree critical point is derived from a parameter-free recursion, and the circuit identification is explicitly conjectural rather than circular.

full rationale

The central exact result, Eq. (12), is obtained from an explicit recursion for the squared minimal singular value Z of random tree tensor networks (Secs. IV E and IV F). The recursion has no free parameters fitted to the target critical point: the identities ⟨Ai⟩ = 1 and ⟨Ai^{1/2} ln Ai⟩ = 0 follow from the U(2) invariance of the node-tensor distribution, and the Haar averages (Eqs. (64) and (65)) are computed analytically. The resulting pc is therefore an independent derivation, not an input. The nonlinear scaling in Eq. (13) is derived from a conjectured but explicitly stated toy-model reduction (Sec. IV H), and the paper openly labels the continuum description a conjecture supported by numerical recursions; this is an acknowledged modeling assumption, not a circular inference. The mapping of the tree critical point to the all-to-all FMPT circuit is explicitly stated as a 'plausible conjecture' (Sec. II D) and as 'far from being a proof' (Sec. IV K), so any failure of that mapping is an unproven assumption rather than a circularity. The all-to-all numerics are presented as consistency checks, with the paper stating that rc 'cannot be pinned down' accurately and that a 'more stringent test of the identity of the two transition points would be valuable.' The field-theory section explicitly disclaims that the Lagrangians are 'conjectures based on symmetry considerations and certain limited consistency checks.' No fitted parameter is renamed as a prediction, and no load-bearing result is justified solely by self-citation: references to the authors' prior work (e.g., Ref. [43] for phase cancellation) provide background and are not used to force the central result. Thus the derivation chain is self-contained, and the main risk is the accuracy of the tree-to-circuit conjecture, not circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The central tree result is parameter-free under its stated statistical assumptions. The jump from tree to circuit, the nonlinear scaling model, and both field theories are explicit conjectures. No new physical particles or forces are introduced; the only invented object is the effective pairing field.

free parameters (2)
  • c0 (time-shift in percolation collapse) = c0 ~ 1.3
    A nonuniversal shift added to the scaling variable (t - c0)/N^{1/5} to improve data collapse in Fig. 5. It does not change exponents or the location of rc.
  • c1, c2 in fitted guide for a(r) = c1 = 39.4, c2 = 3.8
    Parameters in the dashed curve c1 exp(-c2/sqrt(rc-r)) in Fig. 24. The text states these values have no theoretical significance and only indicate a possible functional form.
assumptions (6)
  • standard math The singular-value recursion can be analyzed via directed polymer and traveling wave theory, including the velocity-selection rule for the generating function.
    Used in Sec. IV F-IV H to convert the linear recursion Eq. (61) into the traveling wave speed Eq. (75) and to obtain the exponential scaling near the transition.
  • ad hoc to paper The all-to-all FMPT circuit's critical point coincides with that of the associated tree tensor network.
    Stated in Sec. II D as a 'plausible conjecture' and argued heuristically in Sec. IV K. The numerical simulations cannot verify it.
  • domain assumption Node tensors in the tree are statistically independent, identically distributed, and invariant under U(2) rotations on lower indices.
    Sec. IV B 1. This is what allows the recursion to be written for singular values alone and restricts the treatment to the FMPT rather than the MPT.
  • ad hoc to paper The universal scaling near the tree transition is captured by the simplified nonlinear toy model Eq. (84) and its continuum Fisher-KPP-like limit Eq. (91).
    Sec. IV H 2 states 'we conjecture that the universal properties can be understood in a simpler model'. Numerical evidence is provided, but no proof.
  • domain assumption The replica trick applies with N to 1 for the MPT and N to 0 for the FMPT and RTN, with symmetry group G_N = (S_N x S_N) ⋊ Z_2.
    Sec. VI A-B. This is standard replica machinery, but its use in constructing continuum Landau theories is a conjecture.
  • ad hoc to paper The simplest Landau-Ginsburg Lagrangians L_X and L_Y, truncated at cubic order, are sufficient to describe the transitions.
    Sec. VI E-G explicitly labels these as speculative conjectures and notes that more elaborate continuum descriptions may be required.
invented entities (1)
  • Pairing field X_{ab} (coarse-grained permutation matrix or Edwards-Anderson-like replica overlap)
    purpose: Order parameter for the MPT, FMPT, and RTN field theories; its ordering distinguishes entangled from disentangled phases.
    Introduced in Sec. VI D-E on the basis of replica symmetry and phase cancellation. It has no falsifiable prediction beyond the phase structure it is designed to represent, so no independent evidence is provided.

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Cite this review

Pith. "Pith review of Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in Landau-Ginsburg theory." pith.science (2026). https://pith.science/paper/PISOTJAF

@misc{pith2026200911311,
  author       = {Pith},
  title        = {Pith review of: Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in Landau-Ginsburg theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PISOTJAF}},
  note         = {Machine review of arXiv:2009.11311}
}
read the original abstract

A quantum many-body system whose dynamics includes local measurements at a nonzero rate can be in distinct dynamical phases, with differing entanglement properties. We introduce theoretical approaches to measurement-induced phase transitions (MPT) and also to entanglement transitions in random tensor networks. Many of our results are for "all-to-all" quantum circuits with unitaries and measurements, in which any qubit can couple to any other, and related settings where some of the complications of low-dimensional models are reduced. We also propose field theory descriptions for spatially local systems of any finite dimensionality. To build intuition, we first solve the simplest "minimal cut" toy model for entanglement dynamics in all-to-all circuits, finding scaling forms and exponents within this approximation. We then show that certain all-to-all measurement circuits allow exact results by exploiting local tree-like structure in the circuit geometry. For this reason, we make a detour to give general universal results for entanglement phase transitions random tree tensor networks, making a connection with classical directed polymers on a tree. We then compare these results with numerics in all-to-all circuits, both for the MPT and for the simpler "Forced Measurement Phase Transition" (FMPT). We characterize the two different phases in all-to-all circuits using observables sensitive to the amount of information propagated between initial and final time. We demonstrate signatures of the two phases that can be understood from simple models. Finally we propose Landau-Ginsburg-Wilson-like field theories for the MPT, the FMPT, and entanglement transitions in random tensor networks. This analysis shows a surprising difference between the MPT and the other cases. We discuss measurement dynamics with additional structure (e.g. free-fermion structure), and questions for the future.

Figures

Figures reproduced from arXiv: 2009.11311 by the authors.

Figure 2
Figure 2. FIG. 2. Some of the approaches in this paper. To gain intu [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 1
Figure 1. FIG. 1. Random circuits with measurements. Vertical lines [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The entanglement entropy of states and operators [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (32 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Example of a small unitary circuit with [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: (inset) shows raw data for the percolation probability Pperc of the classical graph (for the all-to-all circuit) as a function of time and N. As can be seen in the main panel, this data collapses onto a single curve when Pperc is plotted against (t + c0)/N1/5 , where c…
Figure 7
Figure 7. Figure 7: FIG. 7. The cost of the minimal cut, [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: (Left) is a schematic of the first k = 3 genera￾tions of the tree that is connected to one end of a link somewhere in the bulk of the circuit. For later conve￾nience we have used a slightly different definition of the tree to that in Sec. III A. Previously we “pruned o…
Figure 9
Figure 9. Figure 9: FIG. 9. The evolution of the typical value of [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: shows vp(λ), defined in Eq. 75, for the Haar tree and the ∆t = 0.3 tree, in the vicinity of their re￾spective pc values. (Numerically, these are obtained by simple averages using a single tensor. In the Haar case, Eqs. 64, 65 also give the exact form.) cp is given by …
Figure 11
Figure 11. Figure 11: FIG. 11. Phase diagram of the quantum tree generated by [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Schematic of a directed polymer on a tree. [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. The distribution of ln [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Critical behaviour for the Haar and ∆ [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Critical behaviour of the classical non-linear tree [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]
Figure 16
Figure 16. Figure 16: shows data for this for the Haar ensemble. The data is compatible with, though it does not clearly establish, the scaling ln Z typ k ∼ k 1/3 which is suggested by the following argument for the continuum model. We expect that for the time-dependent equation xfront dri…
Figure 17
Figure 17. Figure 17: FIG. 17. Main panel: The exponent [PITH_FULL_IMAGE:figures/full_fig_p028_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Schematic: tree tensor network wavefunction for a [PITH_FULL_IMAGE:figures/full_fig_p029_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. The opEE, [PITH_FULL_IMAGE:figures/full_fig_p031_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20. As in Fig [PITH_FULL_IMAGE:figures/full_fig_p032_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21. Heatmap: the squared overlap [PITH_FULL_IMAGE:figures/full_fig_p032_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22. The squared overlap between the states [PITH_FULL_IMAGE:figures/full_fig_p033_22.png]
Figure 23
Figure 23. Figure 23: FIG. 23. The typical distance [PITH_FULL_IMAGE:figures/full_fig_p033_23.png]
Figure 25
Figure 25. Figure 25: FIG. 25. Evidence that the forced measurement (FM) proto [PITH_FULL_IMAGE:figures/full_fig_p034_25.png]
Figure 27
Figure 27. Figure 27: FIG. 27. Schematic: mapping a circuit with unitaries and [PITH_FULL_IMAGE:figures/full_fig_p035_27.png]
Figure 28
Figure 28. Figure 28: FIG. 28. Schematic: boundary conditions in model for effec [PITH_FULL_IMAGE:figures/full_fig_p037_28.png]
Figure 29
Figure 29. Figure 29: FIG. 29. Left: The simple random tensor network described [PITH_FULL_IMAGE:figures/full_fig_p039_29.png]
Figure 30
Figure 30. Figure 30: FIG. 30. At large times, a system in [PITH_FULL_IMAGE:figures/full_fig_p044_30.png]
Figure 31
Figure 31. Figure 31: FIG. 31. The field theory description of the Majorana mea [PITH_FULL_IMAGE:figures/full_fig_p047_31.png]
Figure 32
Figure 32. Figure 32: shows the percolation probability at the crit￾ical point, plotted as a function of time for different sys￾tem sizes N. A good scaling is observed as a function of the variable t/N1/5 , as suggested by Eq. 19. At a fixed value of c and N, the percolation probability Pp…
Figure 33
Figure 33. Figure 33: FIG. 33. The time scale [PITH_FULL_IMAGE:figures/full_fig_p052_33.png]
Figure 36
Figure 36. Figure 36: FIG. 36. An example of the extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p053_36.png]
Figure 38
Figure 38. Figure 38: FIG. 38. The distribution function of the parameters [PITH_FULL_IMAGE:figures/full_fig_p055_38.png]
Figure 39
Figure 39. Figure 39: FIG. 39 [PITH_FULL_IMAGE:figures/full_fig_p059_39.png]
Figure 40
Figure 40. Figure 40: FIG. 40. Same as in Fig [PITH_FULL_IMAGE:figures/full_fig_p059_40.png]
Figure 41
Figure 41. Figure 41: FIG. 41. Fits of ln [PITH_FULL_IMAGE:figures/full_fig_p060_41.png]

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Reference graph

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    Minimal cut formula on tree Assume that the minimal cut chops out m subtrees from the full tree, with each subtree being cut only 59 once, at its apex. The singular values of a given subtree a∈{ 1,...,m } are{λ(a) 1 ,λ (a) 2 }, with (λ(a) 1 )2 + (λ(a) 2 )2 = 1 and (λ(a) 2 )2 =...

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