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Manifolds in high dimensional random landscape: complexity of stationary points and depinning

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Pith's one-line read For a $d<4$ elastic manifold in a high-dimensional random medium, the annealed complexities of stationary points and of minima are explicit functions of the curvature $\mu$, and both vanish at the Larkin mass $\mu_c$.

desk verdict A clean, genuinely new extension of the d=0 and d=1,N=1 complexity results to 1≤d<4, but the central determinant-averaging conjecture is explicitly unproved and the consistency check does not test the variance it would need. read the letter →

arxiv 1908.09217 v2 pith:X6VPIKWM submitted 2019-08-24 cond-mat.dis-nn cond-mat.stat-mechmath-phmath.MP

Yan V Fyodorov , Pierre Le Doussal This is my paper · ORCID
classification cond-mat.dis-nncond-mat.stat-mechmath-phmath.MP
keywords annealedcomplexityrandomenergylandscapeelasticmanifoldKac-RiceformulaLarkinmassdepinningthresholdmatrixspectraldeterminantreplicasymmetrybreaking
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 counts the stationary points and local minima of the energy landscape of an elastic manifold dragged through a high-dimensional random medium. It claims that for internal dimension $d<4$ and large embedding dimension $N$, the annealed complexities—the log of the mean number of equilibria per degree of freedom—are given by explicit formulas: Eq. (32) for all stationary points and Eq. (43) for minima. Both complexities vanish exactly at the curvature $\mu_c$ known as the Larkin mass, so the landscape switches from simple (no exponentially many equilibria) to complex (exponentially many) as $\mu$ drops below $\mu_c$. The vanishing is quadratic in $\mu_c-\mu$ for stationary points and cubic for minima. The same formulas have a finite massless limit, which yields a bound on the force needed to depin the manifold.

What carries the argument

The load-bearing machinery is the Kac-Rice formula, which writes the mean number of stationary points as an integral over configurations of $|\det K_0|$, where $K_0$ is the Hessian. The Hessian has a block structure: off-diagonal blocks come from the discrete Laplacian, while diagonal blocks are independent GOE matrices plus Gaussian shifts $\xi(x)$, so its mean spectral density obeys the self-consistent resolvent equation (19). The decisive step is the conjecture (13), that the GOE average of the absolute determinant can be replaced by the exponential of the average log-determinant; spectral rigidity makes the log-determinant self-averaging. That replacement turns a difficult determinant average into a saddle-point integral over the shifts, whose solution reduces the complexity to integrals involving a single parameter $y(\mu)$ defined by Eq. (33).

What would settle it

Numerically evaluate, for the block-banded Hessian with block size $N$ and $L$ blocks, the ratio $\langle|\det(K+X+\mu I)|\rangle / \exp(\langle\operatorname{Tr}\log|K+X+\mu I|\rangle)$ for increasing $N$ at fixed $L$ and then for increasing $L$. If the ratio does not approach $1$, or if it deviates once $L$ grows before $N\to\infty$, the conjecture (13) fails and the formulas (32) and (43) are not justified.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the annealed complexity of the high-dimensional random-manifold landscape is controlled by the mean spectral density of a block random-matrix Hessian. For curvature $\mu>\mu_c$ the complexity of stationary points and of minima is identically zero. For $\mu<\mu_c$, with $\mu_c$ set by $1=\int_k (\mu_c-t\Delta(k))^{-2}$, the total complexity is $\Sigma(\mu)=\int_{\mu_c}^{\mu} d\tilde\mu \int_k \frac{y(\tilde\mu)^2}{(\tilde\mu-t\Delta(k))((\tilde\mu-t\Delta(k))^2+y(\tilde\mu)^2)}$ and the minima complexity is $\Sigma_{\rm st}(\mu)=-\frac12(\mu_c-\mu)^2+\int_\mu^{\mu_c}(I_1(\tilde\mu)-I_1(\mu_c))\,d\tilde\mu$, with $y(\mu)$ determined by $1=\int_k ((\mu-t\Delta(k))^2+y(\mu)^2)^{-1}$. Near $\mu_c$, $\Sigma\propto(\mu_c-\mu)^2$ while $\Sigma_{\rm st}\propto(\mu_c-\mu)^3$, and the massless limits give the depinning bound $f_c\le f_{\rm st}^c=\sqrt{4B'(0)\Sigma_{\rm st}(0)}$.

Load-bearing premise

Every explicit formula rests on the conjecture that the logarithm of the absolute determinant of the random block Hessian is self-averaging at large $N$, so the average of the determinant is the exponential of the average log-determinant, even when the number of blocks tends to infinity.

Editorial extensions

If this is right

  • For $\mu>\mu_c$ the complexity is zero, so a typical landscape has no exponentially large family of stationary points or minima.
  • For $\mu<\mu_c$ the number of equilibria grows exponentially with $N L^d$ at rate $\Sigma(\mu)$, and the rate for minima vanishes cubically at the transition while the total rate vanishes quadratically.
  • The massless limit yields $N_{\rm tot}\sim e^{C N (L/L_c)^d}$, so the Larkin length $L_c$ controls the exponential count of equilibria in the critical limit.
  • The depinning threshold under a uniform force obeys $f_c\le \sqrt{4B'(0)\Sigma_{\rm st}(0)}$, a bound sharper than the bound obtained from the total stationary-point complexity.
  • In $d=0$ the formulas reproduce the earlier toy-model complexity, and in the continuum the complexity of minima has an elementary closed form for $d<4$, including a finite limit at $d=2$.

Reading between the lines

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

  • If the spectral rigidity conjecture survives the infinite-block limit, the same saddle-point calculation should extend to the quenched complexity, since the log-determinant fluctuations that distinguish annealed from quenched averages are subextensive; the paper leaves that equality open.
  • The universal dimension-dependent ratio $\Sigma_{\rm st}(0)/\Sigma(0)$ (about $0.63$ in $d=1$ and $0.405$ in $d=2$ for the continuum model) predicts a quantitative relation between metastable-state count and the depinning bound that numerical interface simulations could test.
  • A direct testable extension is to count stationary points in finite-size discrete manifolds and check the predicted quadratic and cubic vanishing laws, with finite-$N$ corrections expected from random-matrix edge fluctuations as in the $d=0$ case.
  • The bound $f_c\le f_{\rm st}^c$ may become an equality in models where annealed and quenched complexities coincide, while the random-field-type models cited in the paper suggest settings where the inequality should be strict.
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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 / 4 minor

Summary. The manuscript studies the energy landscape of an elastic manifold with internal dimension d<4 embedded in a high-dimensional random medium, with energy functional (1) and Gaussian disorder covariance (2). Using the Kac-Rice formula, the authors define annealed complexities for the total number of stationary points and for local minima (Eq. (3)). They decompose the Hessian into a block random matrix and, under the spectral-rigidity conjecture stated in Eq. (13) (extended to minima in Eq. (39)), derive explicit formulas for the two complexities: Eqs. (32) and (43). These formulas vanish at the Larkin mass μ_c, with quadratic behavior for stationary points and cubic behavior for minima near μ_c. The massless limit μ→0 is used to obtain an upper bound f_st^c for the depinning threshold, Eq. (49). The supplementary material contains the near-transition expansions, explicit results for the continuum model in d=1,2,3, and a consistency check for the mean resolvent.

Significance. If the central claims hold, the paper provides one of the first explicit high-dimensional landscape-complexity calculations for disordered elastic manifolds with internal dimension d<4, extending the d=0 toy model and connecting landscape topology to depinning. The derivation is transparent and systematic: the saddle-point equations, the self-consistent resolvent equation, and the explicit continuum-model formulas are carefully presented. The authors also give machine-checkable algebraic details in the supplementary material, including exact results in d=2, a numerical constant in d=1, and a universal ratio Σ_st(0)/Σ(0). The conditional nature of the central step is openly acknowledged, which is a strength. However, the main quantitative formulas and the depinning bound are not rigorously established because they depend on an unproved spectral-rigidity conjecture in the infinite-block limit; the consistency check provided tests only the first moment of the log-determinant, not the fluctuations that the conjecture requires to vanish.

major comments (3)
  1. [Eq. (13) and Supplementary Material, section 'LIMIT q→0 AND RESOLVENT'] The replacement ⟨|det(K+X+μI)|⟩ ≈ exp⟨Tr log|K+X+μI|⟩ in Eq. (13) is load-bearing for the central formulas (32) and (43). It is a theorem for L^d=1, but here it must hold after the number of blocks L^d is sent to infinity, not only for finite block number. The supplementary consistency check computes only the q→0 derivative of log|det|, i.e., the first moment of the log-determinant or the mean resolvent. It does not test the variance or higher cumulants. If Var(log|det(K+X+μI)|) grows as c N L^d, or if log⟨|det|⟩−⟨log|det|⟩ is of order N L^d, then the exponent in the annealed complexity acquires an additive correction c/2, which would change the vanishing exponents near μ_c and the value of f_st^c in Eq. (49). The authors should either provide a proof of the required rigidity in the infinite-block limit, or present numerical evidence directly for the fluctuations of the log-determinant, or explicitly restrict the claims in the abstract and conclusions to the conjectural status of this step.
  2. [Eq. (39) and paragraph after Eq. (42)] The complexity of local minima additionally relies on the claim that the integral over the domain D is dominated by the uniform boundary configuration ξ_e = −λ_e^− − μ. This is not automatic: it requires that the constrained saddle point of the action on the boundary of D is the constant configuration, which in turn relies on delocalization of the low-energy eigenmodes of the block banded matrix K. That delocalization is part of the same spectral-rigidity assumption used in Eq. (13). If the boundary saddle were nonuniform, Eq. (43) and the cubic law (45) would not follow. This step should be derived explicitly or stated as a separate conjecture with a numerical test, since the cubic vanishing at μ_c is one of the headline results.
  3. [Eq. (5) and Eq. (49)] The depinning bound f_c ≤ f_st^c = sqrt(4B'(0) Σ_st(μ=0)) is derived from the annealed complexity of minima. As the authors note, the quenched complexity may differ from the annealed one, so the bound is conditional not only on the rigidity conjecture but also on the relation between annealed and quenched counts. This is not an error, but because the bound is one of the two headline results, the conditional status should be stated in the abstract and in the concluding discussion, not only in the body near Eq. (5).
minor comments (4)
  1. [Eq. (43)] There is a parenthesis typo in the displayed formula: the integrand should read (I_1(μ̃) − I_1(μ_c)) dμ̃, not (I_1(μ̃) − I_1(μ_c)dμ̃ as printed.
  2. [After Eq. (34)] The notation I_p(μ) is defined after Eq. (34) but used earlier in Eq. (43) before that definition; please define I_1(μ) before its first use.
  3. [References] Reference [33] is listed as 'in preparation'; if a published or preprint version is available by the time of revision, it should be updated.
  4. [Supplementary Material, Eq. (87)] The numerical constant Σ(μ=0)|_{d=1} ≈ 0.375 t^{-2/3} and the related constants C_{∞,d} in Eq. (91) would benefit from a brief statement of the numerical integration method or an independent check, since these are quoted as decimal values.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity in the complexity derivation; the only self-referential element is a supplementary resolvent consistency check, while the main formulas rest on an explicitly labeled conjecture.

  1. other [Supplementary Material, 'LIMIT q → 0 AND RESOLVENT', Eqs. (96)-(103)]
    "From (19) in the text we see that ir−µ satisfies ... Comparing with (95) we see that ir−µ = ip, hence (102) implies G = −Re(ip), hence (100) leads to the correct result for the real part of the mean resolvant Re G(λ = 0, 0)."

    The check evaluates the q-derivative of log|det| using the paper's conjectured determinant averaging, then evaluates the resulting expression with Eq. (19), the resolvent self-consistent equation. The target Eq. (95) is the same Eq. (19) specialized to λ = −µ. The agreement is therefore by construction: the check verifies only that the determinant formalism is internally consistent with its own input resolvent equation, not that the spectral-rigidity conjecture suppresses determinant fluctuations in the infinite-block limit. Since the main complexity formulas (32)/(43) rely on that conjecture, this check does not independently support them; however the formulas are openly stated as consequences of the conjecture, not as independent predictions.

full rationale

The total and minima complexities (32) and (43) are obtained from the Kac-Rice formula plus an explicitly stated conjecture (13)/(39) that the annealed determinant average is dominated by the exponential of the averaged log. This is an acknowledged assumption, not a disguised fit or a renamed input. The mean resolvent equation (19) used to close the saddle-point calculation is imported from the authors' prior work but is independently supported by the rigorous Khorunzhy-Pastur result [30], so the self-citation is not load-bearing in a circular way. The minima conjecture (39) is supported by spectral-rigidity arguments and an in-preparation self-citation [33]; because the text labels it a conjecture, this is an evidentiary limitation rather than a circular derivation. The depinning bounds (5)/(49) follow by Jensen's inequality from the computed annealed complexity, with no parameter fitted to the target quantity. The only self-referential move is the supplementary q→0 resolvent check, which reproduces Eq. (95) from Eq. (19) by construction and therefore does not provide independent evidence for the conjecture; this is a weakness in the verification chain but does not make the derivation circular. Overall, the paper's central claim is a conditional calculation under a clearly labeled conjecture, with no prediction that reduces to its inputs by definition.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The model parameters (curvature mu0, stiffness t0, disorder amplitude B) are physical inputs, not fitted. No numbers are tuned to data. The central derivation rests on two unproved conjectures about spectral rigidity, on standard saddle-point methods, and on the Kac-Rice formula. No new particles, forces, or dimensions are introduced.

assumptions (4)
  • ad hoc to paper The spectral rigidity of the block banded random matrix K allows the substitution avg(|det(K+X+mu I)|) ≈ exp(avg Tr log|K+X+mu I|) (Eq. 13).
    Explicitly labeled a conjecture in the text. Proved only for L^d = 1 and conjectured for finite block numbers, then extrapolated to an infinite number of blocks.
  • ad hoc to paper For minima, the determinant times the positive-definite step function is dominated by the same saddle point except at the boundary of the domain D, with exponentially small contribution outside D (Eq. 39).
    Conjectured from large deviation principles. The d = 0 version is cited to an unpublished manuscript by Ben Arous, Fyodorov, and Khorzhenko.
  • domain assumption The integral over the Gaussian fields xi(x) is dominated by a uniform saddle point xi(x) = xi* in the complex phase (Eqs. 16-17).
    Standard large-N saddle-point analysis; assumes translational invariance of the solution in the internal space.
  • standard math The Kac-Rice formula (Eq. 7) correctly gives the mean number of stationary points of the energy functional.
    Standard tool in the complexity literature, used in the earlier d = 0 and d = 1, N = 1 works.

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Cite this review

Pith. "Pith review of Manifolds in high dimensional random landscape: complexity of stationary points and depinning." pith.science (2026). https://pith.science/paper/X6VPIKWM

@misc{pith2026190809217,
  author       = {Pith},
  title        = {Pith review of: Manifolds in high dimensional random landscape: complexity of stationary points and depinning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X6VPIKWM}},
  note         = {Machine review of arXiv:1908.09217}
}
abstract

We obtain explicit expressions for the annealed complexities associated respectively with the total number of (i) stationary points and (ii) local minima of the energy landscape for an elastic manifold with internal dimension $d<4$ embedded in a random medium of dimension $N \gg 1$ and confined by a parabolic potential with the curvature parameter $\mu$. These complexities are found to both vanish at the critical value $\mu_c$ identified as the Larkin mass. For $\mu<\mu_c$ the system is in complex phase corresponding to the replica symmetry breaking in its $T=0$ thermodynamics. The complexities vanish respectively quadratically (stationary points) and cubically (minima) at $\mu_c^-$. For $d\geq 1$ they admit a finite "massless" limit $\mu=0$ which is used to provide an upper bound for the depinning threshold under an applied force.

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Works this paper leans on

41 extracted references · 39 canonical work pages

  1. [1]

    Statics and Dynamics of Disordered Elastic Systems

    For review see, Statics and dynamics of disordered elast ic systems, T. Giamarchi, P. Le Doussal. in: “Spin glasses and Random fields”, Series on Directions in condensed matter physics vol 12, Editor A.P. Young, World Scien- tific (1998) [cond-mat/9705096]

    work page Pith review arXiv 1998

  2. [2]

    D. S. Fisher, Sliding charge-density waves as a dynamic critical phenomenon. Phys. Rev. B 31, 1396–1427 (1985)

    work page 1985

  3. [3]

    Giamarchi, P

    T. Giamarchi, P. Le Doussal. Elastic theory of flux lattic es in presence of weak disorder. Phys. Rev. B 52 1242 – 1270 (1995)

    work page 1995

  4. [4]

    Rosso and W

    A. Rosso and W. Krauth, Roughness at the depinning threshold for a long-range elastic string. Phys. Rev. E 65, 025101 (2002)

    work page 2002

  5. [5]

    Le Doussal, K

    P. Le Doussal, K. J. Wiese, P. Chauve, Functional Renor- malization Group and the Field Theory of Disordered Elastic Systems. Phys.Rev.E 69, 026112 (2004)

    work page 2004

  6. [6]

    Le Doussal and K

    P. Le Doussal and K. J. Wiese, A valanche dynamics of elastic interfaces. Phys. Rev. E 88, 022106 (2013)

    work page 2013

  7. [7]

    Mézard and G

    M. Mézard and G. Parisi, Replica field theory for random manifolds. J. Phys. I (France) 1, 809–836 (1991)

    work page 1991

  8. [8]

    Mezard and G

    M. Mezard and G. Parisi. Manifolds in random media: two extreme cases. J.Phys.I France 2, 2231 – 2242 (1992)

    work page 1992

Show all 41 references
  1. [9]

    Balents, J.-P

    L. Balents, J.-P. Bouchaud, and M. Mezard, The large scale energy landscape of randomly pinned objects. J. Phys. I (France) 6, 1007–1020 (1996)

    1996

  2. [10]

    Functional Renor- malization Group at Large N for Disordered Elastic Systems, and Relation to Replica Symmetry Breaking

    Pierre Le Doussal, Kay Joerg Wiese. Functional Renor- malization Group at Large N for Disordered Elastic Systems, and Relation to Replica Symmetry Breaking. Phys.Rev.B 68 174202 (2003)

    2003

  3. [11]

    Le Doussal, M

    P. Le Doussal, M. Mueller, K. J. Wiese. Cusps and shocks in the renormalized potential of glassy random manifolds: How Functional Renormalization Group and Replica Sym- metry Breaking fit together. Phys. Rev. B 77, 064203 (2008) (39 pages)

    2008

  4. [12]

    Auffinger, G

    A. Auffinger, G. Ben Arous, and J. Cerny, Random ma- trices and complexity of spin glasses. Commun. Pure. Appl. Math. 66, 165–201 (2013)

    2013

  5. [13]

    Auffinger and G

    A. Auffinger and G. Ben Arous, Complexity of random smooth functions on the high-dimensional sphere, Ann. Probab. 41(6), 4214–4247 (2013)

    2013

  6. [14]

    Subag and O

    E. Subag and O. Zeitouni, The extremal process of crit- ical points of the pure p-spin spherical spin glass model, Probab. Theory Relat. Fields 168, no.3–4, 773–820 (2017) [arXiv:1509.03098]. 6

    2017 arXiv

  7. [15]

    E. Subag. The complexity of spherical p-spin models in second moment approach. Ann. Probab., 45(5):3385–3450 (2017)

    2017

  8. [16]

    V. Ros, G. Ben Arous, G. Biroli, C. Cammarota. Com- plex energy landscapes in spiked-tensor and simple glassy models: ruggedness, arrangements of local minima and phase transitions. Phys. Rev. X 9, 011003 (2019)

    2019

  9. [17]

    Annibale, A

    A. Annibale, A. Cavagna, I. Giardina, and G. Parisi. Su- persymmetric complexity in the Sherrington-Kirkpatrick model. Phys. Rev. E 68, 061103 (2003)

    2003

  10. [18]

    Fyodorov

    Y.V. Fyodorov. Complexity of Random Energy Land- scapes, Glass Transition, and Absolute Value of the Spectral Determinant of Random Matrices. Phys. Rev. Lett. 92, 240601 (2004); Erratum ibid 93, Issue 14 , 149901(E)(2004)

    2004

  11. [19]

    Parisi, Computing the number of metastable states in infinite-range models

    G. Parisi, Computing the number of metastable states in infinite-range models. in Les Houches summer school, Ses- sion LXXXIII , edited by A. Bovier and et al , volume 83, pages 295–329, Amsterdam, 2005, Elsevier

    2005

  12. [20]

    A. J. Bray and D. S. Dean, Statistics of Critical Points o f Gaussian Fields on Large-Dimensional Spaces. Phys. Rev. Lett. 98, 150201 (2007)

    2007

  13. [21]

    Y. V. Fyodorov and I. Williams, Replica Symmetry Breaking Condition Exposed by Random Matrix Calcula- tion of Landscape Complexity. J. Stat. Phys. 129,no.5–6, 1081–1116 (2007)

    2007

  14. [22]

    Y. V. Fyodorov and C. Nadal, Critical Behavior of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom Distribu- tion, Phys. Rev. Lett. 109, 167203 (2012)

    2012

  15. [23]

    M. R. Douglas, B. Shiffman, and S. Zelditch, Criti- cal Points and supersymmetric vacua, I, Commun. Math. Phys. 252, no.1–3, 325–358 (2004)

    2004

  16. [24]

    Easther, A

    R. Easther, A. H. Guth, and A. Masoumi, Count- ing Vacua in Random Landscapes, preprint hep-th arXiv:1612.05224 (2016)

    2016 arXiv

  17. [25]

    Y. V. Fyodorov, High-Dimensional Random Fields and Random Matrix Theory, Markov Process. Related Fields 21, 483–518 (2015)

    2015

  18. [26]

    Fyodorov, P

    Y.V. Fyodorov, P. Le Doussal, A. Rosso, and C. Texier. Exponential number of equilibria and depinning threshold for a directed polymer in a random potential. Annals of Physics 397, 1–64 (2018)

    2018

  19. [27]

    V Fyodorov and P

    Y. V Fyodorov and P. Le Doussal. Hessian spec- trum at the global minimum of high-dimensional random landscapes. J.Phys. A: Math. Theor. 51, 474002 (2018) [https://doi.org/10.1088/1751-8121/aae74f]

    2018 doi

  20. [28]

    V Fyodorov and P

    Y. V Fyodorov and P. Le Doussal. Manifolds pinned by a high-dimensional random landscape: Hessian at the global energy minimum. [e-preprint arXiv:1903.07159]

    1903 arXiv

  21. [29]

    For N > 1 the Middleton theorems do not hold, so it cannot be excluded that the manifold may get pinned or not depending on the initial condition

  22. [30]

    A. M. Khorunzhy and L. A. Pastur. Limits of Infinite In- teraction Radius, Dimensionality and the Number of Com- ponents for Random Operators with Off-Diagonal Ran- domness. Commun. Math. Phys. 153, 605–646 (1993)

    1993

  23. [31]

    Shcherbina

    T. Shcherbina. Universality of the local regime for the with a finite number of blocks. J.Stat.Phys. 155, Issue 3, 466–499 (2014)

    2014

  24. [32]

    See Supplementary Materials

  25. [33]

    Ben Arous, Y.V

    This generalizes to d > 0 the result proved for d = 0 in G. Ben Arous, Y.V. Fyodorov, B.A. Khorzhenko, in preparation. In that case the domain D is simply ξ > −λ − e − µ

  26. [34]

    [28] for a more general problem which involved a parameter µ eff and is given there in (60)

    The expression for λ − e was analyzed in Ref. [28] for a more general problem which involved a parameter µ eff and is given there in (60). The correspondence between the two problems is obtained by setting µ eff = 0 as can be seen by comparing (19) with the self-consistent equat...

  27. [35]

    Toulouse, M

    G. Toulouse, M. Gabay, T. C. Lubensky and J. Vanni- menus. On the order of the spin glass transitions in mean field theory. J. Physique LETTRES 43 L109 – L113 (1982)

    1982

  28. [36]

    Detcheverry, M

    F. Detcheverry, M. L. Rosinberg, and G. Tarjus, Metastable states and T= 0 hysteresis in the random-field Ising model on random graphs. The European Physical Journal B-Condensed Matter and Complex Systems 44.3 (2005): 327-343, F.J. Perez-Reche, M. L. Rosinberg, and G. Tarjus, N...

    2005

  29. [37]

    EXP ANSION NEAR THE TRANSITION Taking derivatives of Eq. (31) in the Letter we obtain Σ ′′(µ ) = ∂µ ξ∗ + I2(µ ) (50) Let us define for convenience Ipq(µ, y 2) := ∫ k (µ − t∆( k))q ((µ − t∆( k))2 + y2)p/ 2 , I p(µ ) := ∫ k 1 (µ − t∆( k))p (51) The equations (25) of the text whic...

  30. [38]

    Here we analyze the equations (32) and (33) which determine Σ( µ ) as a function of µ in the complex phase µ < µ c

    EXPLICIT FORMULAS FOR THE COMPLEXITY IN THE CONTINUUM MOD EL T otal complexity . Here we analyze the equations (32) and (33) which determine Σ( µ ) as a function of µ in the complex phase µ < µ c. Let us consider the continuum model in dimension d, with ∆( k) = −k2. We restric...

  31. [39]

    Let us consider here the continuum m odel ∆( k) = −k2

    LARKIN LENGTH There are several conventions to define the Larkin length Lc, and they simply differ by some constant prefactors in the weak disorder limit. Let us consider here the continuum m odel ∆( k) = −k2. If we stick to the definition Lc = ( κ 2/R ′′′′(0))1/ 3 given for N = ...

  32. [40]

    Fyodorov, P

    Y.V. Fyodorov, P. Le Doussal, A. Rosso, and C. Texier. Exp onential number of equilibria and depinning threshold for a directed polymer in a random potential. Annals of Physics 397, 1–64 (2018)

    2018

  33. [41]

    Fyodorov, P

    Y. Fyodorov, P. Le Doussal, A. Ossipov, in preparation

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