pith. sign in

arxiv: 2512.01772 · v2 · submitted 2025-12-01 · ❄️ cond-mat.dis-nn · cond-mat.mes-hall

Quantum dynamics of monitored free fermions: Evolution of quantum correlations and scaling at measurement-induced phase transition

Igor Poboiko , Alexander D. Mirlin This is my paper

Pith reviewed 2026-05-17 03:11 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.mes-hall
keywords monitored free fermionsmeasurement-induced phase transitionnonlinear sigma modelquantum correlationslocalization timepurification timecharge sharpening
0
0 comments X p. Extension

The pith

Monitored free fermions evolve quantum correlations via an extended nonlinear sigma-model mapping whose long-time scaling predictions are verified numerically at the measurement-induced transition.

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

The paper extends the mapping of monitored free-fermion dynamics to a nonlinear sigma-model field theory so that it covers finite evolution times and initial states that set different boundary conditions. This extension is used to describe how quantum correlations develop over time from their initial form toward the steady-state form. In the long-time limit the system reduces to a quasi-one-dimensional geometry, and the scaling of the localization time—which is simultaneously the purification time and the charge-sharpening time—is derived analytically. These scaling predictions are confirmed by numerical simulations performed in a two-dimensional model in the vicinity of the measurement-induced phase transition, and the same dynamical method is applied to locate the transition point and extract the correlation-length critical exponent.

Core claim

Extending the nonlinear sigma-model mapping to finite evolution time T and to different classes of initial states with distinct boundary conditions permits an analytical description of the gradual build-up of quantum correlations; in the long-time quasi-one-dimensional regime the localization time scales in a manner whose predictions are fully confirmed by d=2 numerics around the measurement-induced transition, thereby allowing numerical determination of the transition location and the associated correlation-length exponent.

What carries the argument

Nonlinear sigma-model field theory with boundary conditions fixed by finite evolution time T and by the class of initial states.

If this is right

  • Quantum correlations develop gradually with increasing evolution time T from their initial-state values toward the steady-state form.
  • The localization time is simultaneously the purification time and the charge-sharpening time.
  • Analytical scaling predictions for the localization time hold around the measurement-induced phase transition.
  • The dynamical approach determines the location of the measurement-induced transition and the value of the correlation-length critical exponent.

Where Pith is reading between the lines

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

  • The same finite-time extension of the mapping could be tested on initial states with different entanglement properties or on lattices of higher coordination number.
  • The extracted critical exponent may be compared directly with exponents obtained from other monitored many-body systems to test for shared universality.
  • Finite-time corrections to the long-time scaling could be quantified numerically to guide experiments that cannot reach the infinite-time limit.

Load-bearing premise

The mapping to the nonlinear sigma-model field theory remains valid for finite evolution time T and for the different classes of initial states that produce distinct boundary conditions.

What would settle it

A d=2 numerical simulation that finds a localization-time scaling inconsistent with the analytic prediction in the vicinity of the measurement-induced transition would falsify the claimed confirmation.

Figures

Figures reproduced from arXiv: 2512.01772 by Alexander D. Mirlin, Igor Poboiko.

Figure 1
Figure 1. Figure 1: FIG. 1. Boundary conditions for NLSM field theory: [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Evolution of the density correlation function [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Evolution of the variance of charge fluctuation in a subsystem, [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Dynamical scaling analysis of MIPT in a [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Example of long-time quantum dynamics in a [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
read the original abstract

We explore, both analytically and numerically, the quantum dynamics of a many-body free-fermion system subjected to local density measurements. We begin by extending the mapping to the nonlinear sigma-model (NLSM) field theory for the case of finite evolution time $T$ and different classes of initial states, which lead to different NLSM boundary conditions. The analytical formalism is then used to study how quantum correlations gradually develop, with increasing $T$, from those determined by the initial state towards their steady-state form. The analytical results are confirmed by numerical simulations for several types of initial states. We further consider the long-time limit, when the system in $d+1$ space-time dimensions becomes quasi-one-dimensional, and analyze the scaling of the ``localization'' time (which is simultaneously the purification time and the charge-sharpening time for this class of problems). The analytical predictions for scaling properties are fully confirmed by numerical simulations in a $d=2$ model around the measurement-induced phase transition. We use this dynamical approach to determine numerically the measurement-induced transition point and the associated correlation-length critical exponent.

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 extends the nonlinear sigma-model (NLSM) mapping for monitored free-fermion systems to finite evolution time T with initial-state-dependent boundary conditions. It studies the gradual development of quantum correlations from initial-state values toward steady-state form, then analyzes scaling of the localization time (simultaneously the purification and charge-sharpening time) in the long-time quasi-1D regime of a d+1 dimensional system. Analytical scaling predictions are stated to be fully confirmed by numerics in a d=2 model near the measurement-induced phase transition; the same dynamical approach is used to numerically locate the transition point and extract the associated correlation-length critical exponent.

Significance. If the finite-T NLSM extension holds with the stated boundary conditions, the work supplies an analytical route to correlation dynamics and scaling across the measurement-induced transition in free fermions, with the d=2 numerical confirmation of localization-time scaling providing a concrete test. The approach of using purification/charge-sharpening dynamics to extract the critical point and exponent is a useful addition to the literature on monitored systems. The result would be of moderate significance for the field, strengthening field-theoretic understanding while remaining tied to the validity of the mapping.

major comments (2)
  1. [Analytical formalism / NLSM mapping extension] The extension of the NLSM mapping to finite T (with distinct boundary conditions for different initial states) is load-bearing for both the correlation-evolution analysis and the subsequent quasi-1D scaling predictions. The manuscript does not appear to include a direct cross-check comparing NLSM correlators against exact free-fermion numerics at intermediate times to validate the boundary-condition implementation before the long-time scaling is applied.
  2. [Numerical simulations around the transition] The numerical extraction of the measurement-induced transition point and correlation-length exponent in the d=2 model relies on fitting the localization-time scaling to the form derived from the finite-T NLSM. Because the mapping itself is an extension whose accuracy at the relevant intermediate times is not independently verified, the reported critical values risk being influenced by the same assumptions; explicit discussion of fit ranges, error bars, and independence from any auxiliary parameters would be required.
minor comments (2)
  1. [Abstract] The abstract refers to 'several types of initial states' without naming them; a brief parenthetical list would improve readability.
  2. [Figures] In figures presenting numerical confirmation of scaling, include system-size dependence and error bars on the extracted localization times to allow assessment of the quasi-1D regime.

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 major comments point by point below. Where appropriate, we have revised the manuscript to strengthen the validation of the NLSM mapping and the presentation of the numerical analysis.

read point-by-point responses

  1. Referee: The extension of the NLSM mapping to finite T (with distinct boundary conditions for different initial states) is load-bearing for both the correlation-evolution analysis and the subsequent quasi-1D scaling predictions. The manuscript does not appear to include a direct cross-check comparing NLSM correlators against exact free-fermion numerics at intermediate times to validate the boundary-condition implementation before the long-time scaling is applied.

    Authors: We appreciate the referee's emphasis on this validation step. The manuscript reports numerical confirmation of the analytical results for the gradual development of quantum correlations from initial-state values for several classes of initial states. To directly address the request for an explicit cross-check of the finite-T NLSM correlators against exact free-fermion numerics at intermediate times, we have added a dedicated comparison in the revised manuscript (new figure and accompanying text). This comparison demonstrates quantitative agreement between the NLSM predictions and exact numerics for the relevant correlation functions across a range of intermediate evolution times T, thereby validating the boundary-condition implementation prior to the long-time scaling analysis. revision: yes

  2. Referee: The numerical extraction of the measurement-induced transition point and correlation-length exponent in the d=2 model relies on fitting the localization-time scaling to the form derived from the finite-T NLSM. Because the mapping itself is an extension whose accuracy at the relevant intermediate times is not independently verified, the reported critical values risk being influenced by the same assumptions; explicit discussion of fit ranges, error bars, and independence from any auxiliary parameters would be required.

    Authors: We agree that additional transparency in the numerical fitting procedure is important. In the revised manuscript we have expanded the relevant section to include: (i) explicit specification of the fit ranges used for the localization-time scaling, (ii) the error bars obtained from the fits together with the fitting procedure, and (iii) robustness checks demonstrating that the extracted transition point and correlation-length exponent remain stable under variations of auxiliary parameters such as the precise time window, system size, and fitting details. The newly added direct NLSM-versus-exact comparison at intermediate times further supports the accuracy of the mapping in the regime relevant to the scaling analysis. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained with independent numerical checks

full rationale

The paper extends the NLSM mapping analytically to finite T with initial-state-dependent boundary conditions, derives predictions for correlation evolution and quasi-1D scaling of localization/purification/charge-sharpening times, and states that these predictions are confirmed by direct numerical simulations of the monitored free-fermion model in d=2. The same numerics are used to locate the measurement-induced transition point and extract the correlation-length exponent. No step reduces a claimed prediction or result to its inputs by construction: the scaling analysis follows from the field theory but is externally validated rather than fitted or renamed; the transition-point determination is numerical and independent of any NLSM parameters. The extension of the mapping is presented as an analytical step whose outputs are checked against exact evolution, satisfying the criteria for a self-contained derivation without load-bearing self-citation chains or self-definitional loops.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on the validity of the NLSM mapping for finite time and on the quasi-one-dimensional reduction in the long-time limit; no explicit free parameters or invented entities are named in the abstract, but the numerical location of the transition implicitly involves fitting or scanning a measurement strength parameter.

free parameters (1)
  • measurement strength at transition
    The transition point is located numerically; this value is determined from the data rather than derived from first principles.
axioms (2)
  • domain assumption The averaged quantum dynamics of monitored free fermions can be mapped to a nonlinear sigma-model field theory for finite evolution time T and for different initial-state boundary conditions.
    This mapping is the starting point of the analytical formalism described in the abstract.
  • domain assumption In the long-time limit the system in d+1 space-time dimensions becomes quasi-one-dimensional.
    This reduction is used to analyze the scaling of the localization time.

pith-pipeline@v0.9.0 · 5504 in / 1529 out tokens · 25600 ms · 2026-05-17T03:11:21.048230+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages · 1 internal anchor

  1. [1]

    propagate

    (a) RatioT ∗(L, γ)/Lof the purification time scaleT ∗(L, γ)to the system sizeL, as a function of measurement rateγfor system sizes fromL=20 toL=52 (see legend). The crossing point marked with an arrow provides the position of the transition pointγ c. (b) Best single-parameter collapse according to Eq. (28), which allows us to determine the critical measur...

    work page

  2. [2]

    Y. Li, X. Chen, and M. P. A. Fisher, Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B98, 205136 (2018)

    work page 2018

  3. [3]

    Skinner, J

    B. Skinner, J. Ruhman, and A. Nahum, Measurement- induced phase transitions in the dynamics of entangle- ment, Phys. Rev. X9, 031009 (2019)

    work page 2019

  4. [4]

    A. Chan, R. M. Nandkishore, M. Pretko, and G. Smith, Unitary-projective entanglement dynamics, Phys. Rev. B 99, 224307 (2019)

    work page 2019

  5. [5]

    X. Cao, A. Tilloy, and A. De Luca, Entanglement in a fermion chain under continuous monitoring, SciPost Phys.7, 024 (2019)

    work page 2019

  6. [6]

    Szyniszewski, A

    M. Szyniszewski, A. Romito, and H. Schomerus, Entan- glement transition from variable-strength weak measure- ments, Phys. Rev. B100, 064204 (2019)

    work page 2019

  7. [7]

    Y. Li, X. Chen, and M. P. A. Fisher, Measurement- driven entanglement transition in hybrid quantum cir- cuits, Phys. Rev. B100, 134306 (2019)

    work page 2019

  8. [8]

    Y. Bao, S. Choi, and E. Altman, Theory of the phase transition in random unitary circuits with measurements, Phys. Rev. B101, 104301 (2020)

    work page 2020

  9. [9]

    A. C. Potter and R. Vasseur, Entanglement dynamics in hybrid quantum circuits, inEntanglement in Spin Chains: From Theory to Quantum Technology Applica- tions, edited by A. Bayat, S. Bose, and H. Johannes- son (Springer International Publishing, Cham, 2022) pp. 211–249

    work page 2022

  10. [10]

    M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Random quantum circuits, Annu. Rev. Condens. Matter Phys.14, 335 (2023)

    work page 2023

  11. [11]

    C. Noel, P. Niroula, D. Zhu, A. Risinger, L. Egan, D. Biswas, M. Cetina, A. V. Gorshkov, M. J. Gullans, D. A. Huse, and C. Monroe, Measurement-induced quan- tum phases realized in a trapped-ion quantum computer, Nat. Phys.18, 760 (2022)

    work page 2022

  12. [12]

    J. M. Koh, S.-N. Sun, M. Motta, and A. J. Minnich, Measurement-induced entanglement phase transition on a superconducting quantum processor with mid-circuit readout, Nat. Phys.19, 1314 (2023)

    work page 2023

  13. [13]

    J. C. Hokeet al.(Google Quantum AI and Collabora- tors), Measurement-induced entanglement and telepor- tation on a noisy quantum processor, Nature622, 481 (2023)

    work page 2023

  14. [14]

    Agrawal, J

    U. Agrawal, J. Lopez-Piqueres, R. Vasseur, S. Gopalakr- ishnan, and A. C. Potter, Observing quantum measure- ment collapse as a learnability phase transition, Phys. Rev. X14, 041012 (2024)

    work page 2024

  15. [15]

    Turkeshi, A

    X. Turkeshi, A. Biella, R. Fazio, M. Dalmonte, and M. Schir` o, Measurement-induced entanglement transi- tions in the quantum Ising chain: From infinite to zero clicks, Phys. Rev. B103, 224210 (2021)

    work page 2021

  16. [16]

    Alberton, M

    O. Alberton, M. Buchhold, and S. Diehl, Entanglement transition in a monitored free-fermion chain: From ex- tended criticality to area law, Phys. Rev. Lett.126, 170602 (2021)

    work page 2021

  17. [17]

    Buchhold, Y

    M. Buchhold, Y. Minoguchi, A. Altland, and S. Diehl, Ef- fective theory for the measurement-induced phase transi- tion of Dirac fermions, Phys. Rev. X11, 041004 (2021)

    work page 2021

  18. [18]

    Coppola, E

    M. Coppola, E. Tirrito, D. Karevski, and M. Collura, Growth of entanglement entropy under local projective measurements, Phys. Rev. B105, 094303 (2022)

    work page 2022

  19. [19]

    Carollo and V

    F. Carollo and V. Alba, Entangled multiplets and spread- ing of quantum correlations in a continuously monitored tight-binding chain, Phys. Rev. B106, L220304 (2022). 15

    work page 2022

  20. [20]

    Szyniszewski, O

    M. Szyniszewski, O. Lunt, and A. Pal, Disordered moni- tored free fermions, Phys. Rev. B108, 165126 (2023)

    work page 2023

  21. [21]

    Poboiko, P

    I. Poboiko, P. P¨ opperl, I. V. Gornyi, and A. D. Mirlin, Theory of free fermions under random projective mea- surements, Phys. Rev. X13, 041046 (2023)

    work page 2023

  22. [22]

    Poboiko, I

    I. Poboiko, I. V. Gornyi, and A. D. Mirlin, Measurement- induced phase transition for free fermions above one di- mension, Phys. Rev. Lett.132, 110403 (2024)

    work page 2024

  23. [23]

    Chahine and M

    K. Chahine and M. Buchhold, Entanglement phases, lo- calization, and multifractality of monitored free fermions in two dimensions, Phys. Rev. B110, 054313 (2024)

    work page 2024

  24. [24]

    Lumia, E

    L. Lumia, E. Tirrito, R. Fazio, and M. Collura, Measurement-induced transitions beyond Gaussianity: A single particle description, Phys. Rev. Res.6, 023176 (2024)

    work page 2024

  25. [25]

    Starchl, M

    E. Starchl, M. H. Fischer, and L. M. Sieberer, General- ized zeno effect and entanglement dynamics induced by fermion counting, PRX Quantum6, 030302 (2025)

    work page 2025

  26. [26]

    M. Fava, L. Piroli, D. Bernard, and A. Nahum, Monitored fermions with conserved U(1)charge, Phys. Rev. Res.6, 043246 (2024)

    work page 2024

  27. [27]

    M. Fava, L. Piroli, T. Swann, D. Bernard, and A. Nahum, Nonlinear sigma models for monitored dynamics of free fermions, Phys. Rev. X13, 041045 (2023)

    work page 2023

  28. [28]

    Xiao and K

    Z. Xiao and K. Kawabata, Topology of monitored quan- tum dynamics (2025), arXiv:2412.06133 [cond-mat.stat- mech]

    work page arXiv 2025

  29. [29]

    Mochizuki and R

    K. Mochizuki and R. Hamazaki, Measurement-induced spectral transition, Phys. Rev. Lett.134, 010410 (2025)

    work page 2025

  30. [30]

    Oshima, K

    H. Oshima, K. Mochizuki, R. Hamazaki, and Y. Fuji, Topology and spectrum in measurement-induced phase transitions, Phys. Rev. Lett.134, 240401 (2025)

    work page 2025

  31. [31]

    H. Pan, H. Shapourian, and C.-M. Jian, Topological modes in monitored quantum dynamics, Phys. Rev. B 112, 144301 (2025)

    work page 2025

  32. [32]

    Z. Xiao, T. Ohtsuki, and K. Kawabata, Universal stochastic equations of monitored quantum dynamics, Phys. Rev. Lett.134, 140401 (2025)

    work page 2025

  33. [33]

    M. J. Gullans and D. A. Huse, Dynamical purifica- tion phase transition induced by quantum measurements, Phys. Rev. X10, 041020 (2020)

    work page 2020

  34. [34]

    Agrawal, A

    U. Agrawal, A. Zabalo, K. Chen, J. H. Wilson, A. C. Potter, J. H. Pixley, S. Gopalakrishnan, and R. Vasseur, Entanglement and charge-sharpening transitions in U(1) symmetric monitored quantum circuits, Phys. Rev. X12, 041002 (2022)

    work page 2022

  35. [35]

    Zabalo, M

    A. Zabalo, M. J. Gullans, J. H. Wilson, S. Gopalakrish- nan, D. A. Huse, and J. H. Pixley, Critical properties of the measurement-induced transition in random quantum circuits, Phys. Rev. B101, 060301(R) (2020)

    work page 2020

  36. [36]

    O. Lunt, M. Szyniszewski, and A. Pal, Measurement- induced criticality and entanglement clusters: A study of one-dimensional and two-dimensional clifford circuits, Phys. Rev. B104, 155111 (2021)

    work page 2021

  37. [37]

    L´ oio, A

    H. L´ oio, A. De Luca, J. De Nardis, and X. Turkeshi, Purification timescales in monitored fermions, Phys. Rev. B108, L020306 (2023)

    work page 2023

  38. [38]

    B. Fan, C. Yin, and A. M. Garc´ ıa-Garc´ ıa, Entangle- ment dynamics of monitored non-interacting fermions on graphic-processing-units (2025), arXiv:2508.18468 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  39. [39]

    Poboiko, P

    I. Poboiko, P. P¨ opperl, I. V. Gornyi, and A. D. Mirlin, Measurement-induced transitions for interacting fermions, Phys. Rev. B111, 024204 (2025)

    work page 2025

  40. [40]

    3 of Ref

    See Fig. 3 of Ref. [20], which shows the reduction of the effective diffusion constantg(q)=C(q)/∣q∣due to ballis- tic and localization effects in the relevant range ofqfor various values ofγ, includingγ=0.1 used in numerical study in Sec. III B of the present work

    work page

  41. [41]

    Barratt, U

    F. Barratt, U. Agrawal, S. Gopalakrishnan, D. A. Huse, R. Vasseur, and A. C. Potter, Field theory of charge sharpening in symmetric monitored quantum circuits, Phys. Rev. Lett.129, 120604 (2022)

    work page 2022

  42. [42]

    MacKinnon and B

    A. MacKinnon and B. Kramer, One-parameter scaling of localization length and conductance in disordered sys- tems, Phys. Rev. Lett.47, 1546 (1981)

    work page 1981

  43. [43]

    J. L. Pichard and G. Sarma, Finite size scaling approach to anderson localisation, Journal of Physics C: Solid State Physics14, L127 (1981)

    work page 1981

  44. [44]

    Slevin and T

    K. Slevin and T. Ohtsuki, The Anderson transition: Time reversal symmetry and universality, Phys. Rev. Lett.78, 4083 (1997)

    work page 1997

  45. [45]

    Kramer, A

    B. Kramer, A. MacKinnon, T. Ohtsuki, and K. Slevin, Finite size scaling analysis of the anderson transition, In- ternational Journal of Modern Physics B24, 1841 (2010), https://doi.org/10.1142/S0217979210064630

    work page doi:10.1142/s0217979210064630 2010

  46. [46]

    Slevin and T

    K. Slevin and T. Ohtsuki, Critical exponent for the an- derson transition in the three-dimensional orthogonal universality class, New Journal of Physics16, 015012 (2014)

    work page 2014

  47. [47]

    Poboiko, I

    I. Poboiko, I. V. Gornyi, and A. D. Mirlin, Mesoscopic fluctuations and multifractality at and across measurement-induced phase transition (2025), arXiv:2507.11312 [quant-ph]

    work page arXiv 2025

  48. [48]

    Evers and A

    F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys.80, 1355 (2008)

    work page 2008

  49. [49]

    Slevin and T

    K. Slevin and T. Ohtsuki, Corrections to scaling at the anderson transition, Phys. Rev. Lett.82, 382 (1999)

    work page 1999

  50. [50]

    Slevin and T

    K. Slevin and T. Ohtsuki, Critical exponent for the quan- tum hall transition, Phys. Rev. B80, 041304 (2009)

    work page 2009

  51. [51]

    Poboiko, M

    I. Poboiko, M. Szyniszewski, C. J. Turner, I. V. Gornyi, A. D. Mirlin, and A. Pal, Measurement-induced l´ evy flights of quantum information, Phys. Rev. Lett.135, 170403 (2025)

    work page 2025

  52. [52]

    H. Guo, M. S. Foster, C.-M. Jian, and A. W. W. Ludwig, Field theory of monitored interacting fermion dynam- ics with charge conservation, Phys. Rev. B112, 064304 (2025)

    work page 2025

  53. [53]

    Kamenev and A

    A. Kamenev and A. Levchenko, Keldysh technique and non-linearσ-model: basic principles and applications, Advances in Physics58, 197 (2009)

    work page 2009

  54. [54]

    Politis, J

    D. Politis, J. Romano, and M. Wolf,Subsampling, Springer Series in Statistics (Springer New York, 1999)

    work page 1999