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The extremal process of critical points of the pure p-spin spherical spin glass model

Abstract

Recently, sharp results concerning the critical points of the Hamiltonian of the p-spin spherical spin glass model have been obtained by means of moments computations. In particular, these moments computations allow for the evaluation of the leading term of the ground-state, i.e., of the global minimum. In this paper, we study the extremal point process of critical points—that is, the point process associated to all critical values in the vicinity of the ground-state. We show that the latter converges in distribution to a Poisson point process of exponential intensity. In particular, we identify the correct centering of the ground-state and prove the convergence in distribution of the centered minimum to a (minus) Gumbel variable. These results are identical to what one obtains for a sequence of i.i.d variables, correctly normalized; namely, we show that the model is in the universality class of REM.

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Notes

  1. 1.

    A decorated Poisson point process is a process whose law is obtained from a Poisson point process by replacing each of the atoms by an independent copy of some point process (called the “decoration”). An SDPPP is simply a decorated Poisson point process, shifted by an independent random variable.

  2. 2.

    For fixed open \(J_N=J\), by the Portmanteau theorem \(\liminf _{N\rightarrow \infty } \mathbb {E}\{ \left( \xi _{N}\left( J\right) \right) ^{2}\}\ge \mathbb {E}\{ \left( \xi _{\infty }\left( J\right) \right) ^{2}\} \) where \(\xi _{\infty }\) is the limiting process of Theorem 1. Proposition 3 (and e.g. [47, Theorem 4.2]) implies that \(\lim _{N\rightarrow \infty } \mathbb {E}\left\{ \xi _{N}\left( J\right) \right\} = \mathbb {E}\left\{ \xi _{\infty }\left( J\right) \right\} \). Therefore, for such \(J_N=J\), it in fact follows from the convergence to Poisson process of Theorem 1 that (1.8) below holds with lim instead of limsup and with equality. We note that the proof of Theorem 1 only requires the upper bound stated in Proposition 4.

  3. 3.

    As we shall see, the only critical points that will be relevant are minimum points.

  4. 4.

    See [47, Theorem 4.2], and note that \(A_{n}\) are continuity sets of \(\bar{\xi }_{\infty }\).

  5. 5.

    This set almost surely contains exactly half of the points in \(\mathscr {C}_{N}\left( L\right) \), and by definition does not contain antipodal points.

  6. 6.

    Up to the negligible event that there exists \(\varvec{\sigma }\in \mathscr {C}_{N}\left( L\right) \) with \(\left\langle \varvec{\sigma },\mathbf {n}\right\rangle =0\).

  7. 7.

    At least when \(\nabla ^{2}\bar{f}_{\varvec{\sigma }}\left( 0\right) \) is invertible, which is the case when \(\mathcal {A}_{1}\) occurs, and we shall indeed restrict to this event when discussing \(Y_{\varvec{\sigma }}\) below.

  8. 8.

    This can be seen by the following. Letting \(\left\{ \tfrac{\partial }{\partial x_{i}}\right\} _{i=1}^{N-1}\) denote the pushforward of \(\left\{ \tfrac{d}{d x_{i}}\right\} _{i=1}^{N-1}\) by P we have that at the north pole, \(\left\{ \tfrac{\partial }{\partial x_{i}} (\mathbf {n})\right\} _{i=1}^{N-1}\) is an orthonormal frame. For any point \(\varvec{\sigma }\) in a small neighborhood of \(\mathbf {n}\) we can define an orthonormal frame as the parallel transport of \(\left\{ \tfrac{\partial }{\partial x_{i}}(\mathbf {n})\right\} _{i=1}^{N-1}\) along the geodesic connecting \(\mathbf {n}\) and \(\varvec{\sigma }\). This yields a smooth orthonormal frame field on this neighborhood, say \(E_i (\varvec{\sigma })=\sum _{j=1}^{N-1}a_{ij} (\varvec{\sigma })\frac{\partial }{\partial x_{j}}(\varvec{\sigma })\), \(i=1,...,N-1\). Working with the coordinate system P one can verify that at \(x=0\) the Christoffel symbols \(\Gamma _{ij}^k\) are equal to 0, and therefore (see e.g. [43, Eq. (2), p. 53]) the derivatives \(\frac{d}{dx_k}a_{ij}(P(x))\) at \(x=0\) are also equal to 0.

  9. 9.

    We note that the integrand in (7.44) is a continuous Radon-Nikodym derivative (as seen from applying the Kac-Rice formula [2, Theorem 12.1.1] to express the mean number of points as above in a subset of the sphere) and therefore it is independent, at each point \(\varvec{\sigma }\), of the choice of the orthonormal frame field \((E_i(\varvec{\sigma }))_{i=1}^{N-1}\).

  10. 10.

    For the cases \(i=2,3,4\) we have continuity and finite variance conditional on \(g_{1}\left( \varvec{\sigma }\right) \in B_{1}\) (and not in general), which is, of course, sufficient since we anyway work under this conditioning and since by continuity if \(g_{1}\left( \varvec{\sigma }_{0}\right) \in B_{1}\) for a particular point \(\varvec{\sigma }_{0}\), then there exists a neighborhood of \(\varvec{\sigma }_{0}\) on the sphere on which \(g_{1}\left( \varvec{\sigma }\right) \in B_{1}\).

References

  1. 1.

    Addario-Berry, L., Reed, B.: Minima in branching random walks. Ann. Probab. 37(3), 1044–1079 (2009)

  2. 2.

    Adler, R.J., Taylor, J.E.: Random fields and geometry. Springer monographs in mathematics. Springer, New York (2007)

  3. 3.

    Aïdékon, E.: Convergence in law of the minimum of a branching random walk. Ann. Probab. 41(3A), 1362–1426 (2013)

  4. 4.

    Aïdékon, E., Berestycki, J., Brunet, É., Shi, Z.: Branching Brownian motion seen from its tip. Probab. Theory Relat. Fields 157(1–2), 405–451 (2013)

  5. 5.

    Aizenman, M., Contucci, P.: On the stability of the quenched state in mean-field spin-glass models. J. Stat. Phys. 92(5), 765–783 (1998)

  6. 6.

    Anderson, G.W., Guionnet, A., Zeitouni, O.: An introduction to random matrices. Cambridge studies in advanced mathematics, vol. 118. Cambridge University Press, Cambridge (2010)

  7. 7.

    Anderson, T.W.: The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Am. Math. Soc. 6, 170–176 (1955)

  8. 8.

    Arguin, L.-P.: Competing particle systems and the Ghirlanda-Guerra identities. Electron. J. Probab. 13(69), 2101–2117 (2008)

  9. 9.

    Arguin, L.-P., Aizenman, M.: On the structure of quasi-stationary competing particle systems. Ann. Probab. 37(3), 1080–1113 (2009)

  10. 10.

    Arguin, L.-P., Bovier, A., Kistler, N.: Genealogy of extremal particles of branching Brownian motion. Comm. Pure Appl. Math. 64(12), 1647–1676 (2011)

  11. 11.

    Arguin, L.-P., Bovier, A., Kistler, N.: The extremal process of branching Brownian motion. Probab. Theory Relat. Fields 157(3–4), 535–574 (2013)

  12. 12.

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

  13. 13.

    Auffinger, A., Ben Arous, G., Černý, J.: Random matrices and complexity of spin glasses. Commun. Pure Appl. Math. 66(2), 165–201 (2013)

  14. 14.

    Bachmann, M.: Limit theorems for the minimal position in a branching random walk with independent logconcave displacements. Adv. Appl. Probab. 32(1), 159–176 (2000)

  15. 15.

    Biskup, M., Louidor, O.: Extreme local extrema of two-dimensional discrete Gaussian free field. Comm. Math. Phys. 345(1), 271–304 (2016)

  16. 16.

    Bolthausen, E.: Random media and spin glasses: an introduction into some mathematical results and problems. In: Spin glasses, volume 1900 of Lecture Notes in Math., pp 1–44. Springer, Berlin (2007)

  17. 17.

    Bolthausen, E., Deuschel, J.-D., Giacomin, G.: Entropic repulsion and the maximum of the two-dimensional harmonic crystal. Ann. Probab. 29(4), 1670–1692 (2001)

  18. 18.

    Bolthausen, E., Deuschel, J.D., Zeitouni, O.: Recursions and tightness for the maximum of the discrete, two dimensional Gaussian free field. Electron. Commun. Probab. 16, 114–119 (2011)

  19. 19.

    Bolthausen, E., Sznitman, A.-S.: On Ruelle’s probability cascades and an abstract cavity method. Commun. Math. Phys. 197(2), 247–276 (1998)

  20. 20.

    Bovier A.: From spin glasses to branching Brownian motion-and back? In: Biskup, M., Cerny, J., Kotecky, R. (eds), Random Walks, Random Fields, and Disordered Systems (Proceedings of the 2013 Prague Summer School on Mathematical Statistical Physics), number 2144 in Lecture Notes in Mathematics 2144. Springer (2015)

  21. 21.

    Bovier, A., Kurkova, I.: Derrida’s generalised random energy models. I. Models with finitely many hierarchies. Ann. Inst. H. Poincaré Probab. Stat. 40(4), 439–480 (2004)

  22. 22.

    Bovier, A., Kurkova, I.: Derrida’s generalized random energy models. II. Models with continuous hierarchies. Ann. Inst. H. Poincaré Probab. Stat. 40(4), 481–495 (2004)

  23. 23.

    Bovier, A., Kurkova, I.: Much ado about Derrida’s GREM. In Spin glasses, volume 1900 of Lecture Notes in Math., pp. 81–115. Springer, Berlin (2007)

  24. 24.

    Bramson, M.: Maximal displacement of branching Brownian motion. Commun. Pure Appl. Math. 31(5), 531–581 (1978)

  25. 25.

    Bramson, M.: Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Am. Math. Soc. 44(285), iv+190 (1983)

  26. 26.

    Bramson, M., Ding, J., Zeitouni, O.: Convergence in law of the maximum of the two-dimensional discrete Gaussian free field. Commun. Pure Appl. Math. 69(1), 62–123 (2016)

  27. 27.

    Bramson, M., Zeitouni, O.: Tightness for a family of recursion equations. Ann. Probab. 37(2), 615–653 (2009)

  28. 28.

    Bramson, M., Zeitouni, O.: Tightness of the recentered maximum of the two-dimensional discrete Gaussian free field. Commun. Pure Appl. Math. 65(1), 1–20 (2012)

  29. 29.

    Carpentier, D., Le Doussal, P.: Glass transition of a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in liouville and sinh-gordon models. Phys. Rev. E 63, 026110 (2001)

  30. 30.

    Chen, W.-K.: The Aizenman-Sims-Starr scheme and Parisi formula for mixed \(p\)-spin spherical models. Electron. J. Probab. 18(94), 14 (2013)

  31. 31.

    Chiarini, A., Cipriani, A., Hazra, R.: A note on the extremal process of the supercritical Gaussian free field. Electron. Commun. Probab. 20(74), 1–10 (2015)

  32. 32.

    Crisanti, A., Sommers, H.-J.: The spherical p-spin interaction spin glass model: the statics. Zeitschrift für Physik B Condensed Matter 87(3), 341–354 (1992)

  33. 33.

    Daley, D.J., Vere-Jones, D.: An introduction to the theory of point processes. Probability and its Applications (New York), vol. II, 2nd edn. Springer, New York (2008). General theory and structure

  34. 34.

    Daviaud, O.: Extremes of the discrete two-dimensional Gaussian free field. Ann. Probab. 34(3), 962–986 (2006)

  35. 35.

    Deift, P., Gioev, D.: Universality in random matrix theory for orthogonal and symplectic ensembles. Int. Math. Res. Pap. IMRP, (2):Art. ID rpm004, 116 (2007)

  36. 36.

    Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math. 52(12), 1491–1552 (1999)

  37. 37.

    Deny, J.: Sur l’équation de convolution \(\mu = \mu \star \sigma \). Séminaire Brelot-Choquet-Deny. Théorie du potentiel 4, 1–11 (1960)

  38. 38.

    Derrida, B.: Random-energy model: an exactly solvable model of disordered systems. Phys. Rev. B (3) 24(5), 2613–2626 (1981)

  39. 39.

    Derrida, B.: A generalization of the random energy model which includes correlations between energies. J. Physique Lett. 46(9), 401–407 (1985)

  40. 40.

    Ding, J.: Exponential and double exponential tails for maximum of two-dimensional discrete Gaussian free field. Probab. Theory Relat. Fields 157(1–2), 285–299 (2013)

  41. 41.

    Ding, J., Roy, R., Zeitouni, O.: Convergence of the centered maximum of log-correlated Gaussian fields. arXiv:1503.04588 (2015)

  42. 42.

    Ding, J., Zeitouni, O.: Extreme values for two-dimensional discrete Gaussian free field. Ann. Probab. 42(4), 1480–1515 (2014)

  43. 43.

    do Carmo, M.P.: Riemannian geometry. Mathematics: theory & applications. Birkhäuser Boston, Inc., Boston, MA, (1992) Translated from the second Portuguese edition by Francis Flaherty

  44. 44.

    Faraut, J.: Logarithmic potential theory, orthogonal polynomials, and random matrices. In: Modern methods in multivariate statistics, Lecture Notes of CIMPA-FECYT-UNESCO-ANR. Hermann (2014)

  45. 45.

    Ghirlanda, S., Guerra, F.: General properties of overlap probability distributions in disordered spin systems. towards parisi ultrametricity. J. Phys. A 31(46), 9149 (1998)

  46. 46.

    Hu, Y., Shi, Z.: Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37(2), 742–789 (2009)

  47. 47.

    Kallenberg, O.: Random measures, 3rd edn. Akademie-Verlag, Berlin (1983)

  48. 48.

    Kistler, N.: Derrida’s random energy models. from spin glasses to the extremes of correlated random fields. In: Gayrard, V., Kistler, N. (eds) Correlated random systems: five different methods, volume 2143 of Lecture Notes in Mathematics. Springer (2015)

  49. 49.

    Lalley, S.P., Sellke, T.: A conditional limit theorem for the frontier of a branching Brownian motion. Ann. Probab. 15(3), 1052–1061 (1987)

  50. 50.

    Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and related properties of random sequences and processes. Springer Series in Statistics. Springer-Verlag, New York-Berlin (1983)

  51. 51.

    Liggett, T.M.: Random invariant measures for Markov chains, and independent particle systems. Z. Wahrsch. Verw. Gebiete 45(4), 297–313 (1978)

  52. 52.

    Madaule, T.: Convergence in law for the branching random walk seen from its tip. preprint. arXiv:1107.2543 [math.PR] (2011)

  53. 53.

    Madaule, T.: Maximum of a log-correlated Gaussian field. Ann. Inst. H. Poincaré Probab. Statist. 51(4), 1369–1431 (2015)

  54. 54.

    Maillard, P.: A note on stable point processes occurring in branching Brownian motion. Electron. Commun. Probab. 18(5), 9 (2013)

  55. 55.

    McDiarmid, C.: Minimal positions in a branching random walk. Ann. Appl. Probab. 5(1), 128–139 (1995)

  56. 56.

    McKean, H.P.: Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Comm. Pure Appl. Math. 28(3), 323–331 (1975)

  57. 57.

    Panchenko, D.: A unified stability property in spin glasses. Commun Math Phys 313(3), 781–790 (2012)

  58. 58.

    Parisi, G.: A sequence of approximated solutions to the S-K model for spin glasses. J Phys A Math Gen. 13(4), L115 (1980)

  59. 59.

    Pickands III, J.: The two-dimensional Poisson process and extremal processes. J. Appl. Probab. 8, 745–756 (1971)

  60. 60.

    Ruzmaikina, A., Aizenman, M.: Characterization of invariant measures at the leading edge for competing particle systems. Ann. Probab. 33(1), 82–113 (2005)

  61. 61.

    Subag, E.: The complexity of spherical \(p\)-spin models - a second moment approach. Ann. Probab. arXiv:1504.02251 (2015) (to appear)

  62. 62.

    Subag, E.: The geometry of the Gibbs measure of pure spherical spin glasses. arXiv:1604.00679 (2016)

  63. 63.

    Subag, E., Zeitouni, O.: Freezing and decorated Poisson point processes. Commun. Math. Phys. 337(1), 55–92 (2015)

  64. 64.

    Talagrand, M.: Free energy of the spherical mean field model. Probab. Theory Relat. Fields 134(3), 339–382 (2006)

  65. 65.

    Wigner, E.P.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 2(62), 548–564 (1955)

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Author information

Correspondence to Eliran Subag.

Additional information

E. S. acknowledges the support of the Adams Fellowship Program of the Israel Academy of Sciences and Humanities. The work of both authors was partially supported by a US-Israel BSF grant and by a grant from the Israel Science Foundation.

Appendices

Appendix 1: proof of Lemma 7

First we remark that by conditioning on \(\eta \), for any \(x\in \mathbb {R}\),

$$\begin{aligned} \mathbb {E}\left\{ \eta _{\infty }^{+}\left( (-\infty ,x)\right) \right\} \le \mathbb {E} \Big \{\sum _i v(x-\eta _i)\Big \} = \int _{\mathbb {R}}e^{ay} v(x-y) dy, \end{aligned}$$

which is finite due to our assumption on v(x); therefore \(\eta _{\infty }^{+}\) is locally finite.

Let \(g:\,\mathbb {R}\rightarrow \mathbb {R}\) be an arbitrary compactly supported, non-negative function which will be fixed throughout the proof. Let \(\kappa >0\) be a large enough constant such that the support of g is contained in \(\left[ -\kappa ,\kappa \right] \). Denote the event

$$\begin{aligned} B=B_{L,N,\kappa }:=\left\{ \left. \eta _{N}^{+}\right| _{\left[ -\kappa ,\kappa \right] }=\left. \bar{\eta }_{N,L}\right| _{\left[ -\kappa ,\kappa \right] }\right\} . \end{aligned}$$

Since this is the same event as in (6.1), defining \(\epsilon (L)\) by

$$\begin{aligned} \liminf _{N\rightarrow \infty }\mathbb {P}\left\{ B_{L,N,\kappa }\right\} =1-\epsilon \left( L\right) , \end{aligned}$$
(7.39)

we have that \(\epsilon \left( L\right) \rightarrow 0\). Denote \(\left\langle g,\zeta \right\rangle \triangleq \int gd\zeta \) and let \(\mathcal {L}_{\zeta }\left[ g\right] \triangleq \mathbb {E}\left\{ \exp \left\{ -\left\langle g,\zeta \right\rangle \right\} \right\} \) be the Laplace functional of \(\zeta \). Then

$$\begin{aligned} \limsup _{N\rightarrow \infty }\left| \mathcal {L}_{\eta _{N}^{+}}\left[ g\right] -\mathbb {E}\left\{ \mathbf {1}_{B}\exp \left\{ -\left\langle g,\eta _{N}^{+}\right\rangle \right\} \right\} \right|= & {} \limsup _{N\rightarrow \infty }\mathbb {E}\left\{ \mathbf {1}_{B^{c}}\exp \left\{ -\left\langle g,\eta _{N}^{+}\right\rangle \right\} \right\} \nonumber \\&\le \epsilon \left( L\right) . \end{aligned}$$
(7.40)

Fix some \(L>0\), let \(\delta >0\), and let \(m_{0}:=m_{0}\left( L,\delta \right) \) be a natural number such that \(\mathbb {P}\left\{ D_{L,N,\delta }\right\} >1-\delta \), for all N, with

$$\begin{aligned} D:=D_{L,N,\delta }\triangleq \left\{ Q_{N,L}\le m_{0}\right\} , \end{aligned}$$

where \(Q_{N,L}=\eta _{N}\left( \left[ -L,L\right] \right) \) as defined in point (1) of the lemma. The existence of \(m_{0}\) is guaranteed by tightness of the convergent sequence \(\eta _{N}\). We have that

$$\begin{aligned} \left| \mathbb {E}\left\{ \mathbf {1}_{B}\exp \left\{ -\left\langle g,\eta _{N}^{+}\right\rangle \right\} \right\} -\mathbb {E}\left\{ \mathbf {1}_{B\cap D}\exp \left\{ -\left\langle g,\eta _{N}^{+}\right\rangle \right\} \right\} \right| \le \delta . \end{aligned}$$
(7.41)

Recall that we assumed in point (1) that \(\left. \eta _{N}\right| _{\left[ -L,L\right] }=\sum _{i=1}^{Q_{N,L}}\delta _{\eta _{N,L,i}}\). With \(\hat{\eta }_{N,L}\triangleq \sum _{i=1}^{Q_{N,L}}\delta _{\eta _{N,L,i}+X_{i}}\),

$$\begin{aligned} \mathbf {1}_{B\cap D}\left| \left\langle g,\eta _{N}^{+}\right\rangle -\left\langle g,\,\hat{\eta }_{N,L}\right\rangle \right| \end{aligned}$$

is stochastically dominated by

$$\begin{aligned} \mathbf {1}_{B\cap D}\sum _{i=1}^{m_{0}}\omega _{g}\left( \left| \bar{X}_{N,i}\left( L\right) -X_{i}\right| \right) , \end{aligned}$$
(7.42)

where \(\omega _{g}\left( t\right) =\sup _{\left| x-y\right| \le t}\left| g\left( x\right) -g\left( y\right) \right| \) is the modulus of continuity of g. Since g is uniformly continuous, \(\omega _{g}\) is continuous. Combining this with point (2) of the lemma implies that (7.42) converges in probability to 0. Therefore, since \(e^{-x}\) is bounded and continuous for \(x\ge 0\),

$$\begin{aligned} \limsup _{N\rightarrow \infty }\left| \mathbb {E}\left\{ \mathbf {1}_{B\cap D}\exp \left\{ -\left\langle g,\eta _{N}^{+}\right\rangle \right\} \right\} -\mathbb {E}\left\{ \mathbf {1}_{B\cap D}\exp \left\{ -\left\langle g,\hat{\eta }_{N,L}\right\rangle \right\} \right\} \right| =0, \end{aligned}$$

and

$$\begin{aligned} \limsup _{N\rightarrow \infty }\left| \mathbb {E}\left\{ \mathbf {1}_{B\cap D}\exp \left\{ -\left\langle g,\eta _{N}^{+}\right\rangle \right\} \right\} -\mathcal {L}_{\hat{\eta }_{N,L}}\left[ g\right] \right| <1-\liminf _{N\rightarrow \infty }\mathbb {P}\left\{ B_{L,N,\kappa }\cap D_{L,N,\delta }\right\} . \end{aligned}$$

Combining this with the bounds we have on \(\mathbb {P}\left\{ D_{L,N,\delta }\right\} \) and \(\mathbb {P}\left\{ B_{L,N,\kappa }\right\} \), (7.41), and (7.40) yields, upon letting \(\delta \rightarrow 0\),

$$\begin{aligned} \lim _{L\rightarrow \infty }\limsup _{N\rightarrow \infty }\left| \mathcal {L}_{\eta _{N}^{+}} \left[ g\right] -\mathcal {L}_{\hat{\eta }_{N,L}}\left[ g\right] \right| =0. \end{aligned}$$

Thus, in order to complete the proof it is sufficient to show that for some sequence \(L_{k}>0\) such that \(\lim _{k\rightarrow \infty }L_{k}=\infty \),

$$\begin{aligned} \lim _{k\rightarrow \infty }\lim _{N\rightarrow \infty }\mathcal {L}_{\hat{\eta }_{N,L_{k}}} \left[ g\right] =\mathcal {L}_{\eta _{\infty }^{+}}\left[ g\right] . \end{aligned}$$
(7.43)

By [33, Lemma 11.1.II], for all but a countable set of values \(L>0\), the interval \(\left[ -L,L\right] \) is a stochastic continuity set of \(\eta \). Thus we can choose \(L_{k}\) as above such that, from [47, Theorem 4.2],

$$\begin{aligned} \left. \eta _{N}\right| _{\left[ -L_{k},L_{k}\right] }\mathop {\mathop {\rightarrow }\limits _{N\rightarrow \infty }}\limits ^{d}\left. \eta \right| _{\left[ -L_{k},L_{k}\right] }. \end{aligned}$$

Hence also

$$\begin{aligned} \hat{\eta }_{N,L_{k}}\mathop {\mathop {\rightarrow }\limits _{N\rightarrow \infty }}\limits ^{d} \sum _{i:\eta _{i}\in \left[ -L_{k},L_{k}\right] }\delta _{\eta _{i}+X_{i}}, \end{aligned}$$

from which (7.43) follows. \(\square \)

Appendix 2: proof of Lemma 13

We recall that \(f(\varvec{\sigma })=f_{N}(\varvec{\sigma })\) is the unit variance random field on \(\mathbb {S}^{N-1}\triangleq \left\{ \varvec{\sigma }\in \mathbb {R}^{N}:\,\left\| \varvec{\sigma }\right\| _{2}=1\right\} \) given by \(f\left( \varvec{\sigma }\right) =\frac{1}{\sqrt{N}}H_{N}(\sqrt{N}\varvec{\sigma })\), \((E_i(\varvec{\sigma }))_{i=1}^{N-1}\) is an arbitrary piecewise smooth orthonormal frame field on the sphere (w.r.t the standard Riemannian metric), and

$$\begin{aligned} \nabla f\left( \varvec{\sigma }\right) =\left( E_{i}f\left( \varvec{\sigma }\right) \right) _{i=1}^{N-1},\,\,\nabla ^{2}f\left( \varvec{\sigma }\right) =\left( E_{i}E_{j}f\left( \varvec{\sigma }\right) \right) _{i,j=1}^{N-1}. \end{aligned}$$

Provided that certain regularity conditions hold, the Kac-Rice formula [2, Theorem 12.1.1] expresses the mean number of points \(\varvec{\sigma }_{0}\) on the sphere at which \(\nabla f\left( \varvec{\sigma }_{0}\right) =0\), \(\sqrt{N} f\left( \varvec{\sigma }_{0}\right) \in (m_N-L,m_N+L)\), and \(g(\varvec{\sigma }_0)\in B'\), for some random field \(g(\varvec{\sigma })\). For this one needs to apply [2, Theorem 12.1.1] with f in the statement of [2, Theorem 12.1.1] being equal to ‘our’ \(\nabla f\left( \varvec{\sigma }\right) \), and with \(h(\varvec{\sigma })=(\sqrt{N} f\left( \varvec{\sigma }\right) ,g(\varvec{\sigma }))\) and \(B=(m_N-L,m_N+L)\times B'\). This yields

$$\begin{aligned}&\mathbb {E}\left\{ \#\left\{ \sqrt{N}\varvec{\sigma }\in \mathscr {C}_{N} \left( L\right) :\,g(\varvec{\sigma })\in B' \right\} \right\} \nonumber \\&\quad =\mathbb {E}\left\{ \#\left\{ \varvec{\sigma }\in \mathbb {S}^{N-1}: \nabla f\left( \varvec{\sigma }\right) =0,\,h\left( \varvec{\sigma }\right) \in B \right\} \right\} = \int _{\mathbb {S}^{N-1}}d\mu (\varvec{\sigma })\varphi _{\nabla f \left( \varvec{\sigma }\right) }\left( 0\right) \nonumber \\&\qquad \times \mathbb {E}\left\{ \left| \det \left( \nabla ^{2}f \left( \varvec{\sigma }\right) \right) \right| \mathbf {1}\Big \{\left| \sqrt{N}f \left( \varvec{\sigma }\right) -m_{N}\right| <L,\,g\left( \varvec{\sigma }\right) \in B'\Big \}\,\Bigg |\,\nabla f\left( \varvec{\sigma }\right) =0\right\} ,\nonumber \\ \end{aligned}$$
(7.44)

where \(\mu \) is the standard (Hausdorff) measure on the sphere and \(\varphi _{\nabla f\left( \varvec{\sigma }\right) }\left( 0\right) \) is the Gaussian density of \(\nabla f\left( \varvec{\sigma }\right) \) at 0. Assuming that \((f(\varvec{\sigma }),g(\varvec{\sigma }))\) is a stationary field on the sphere, one can replace \(\varvec{\sigma }\) everywhere in the right-hand side above by \(\mathbf {n}\), remove the integration, and multiply the right-hand side above by \(\omega _{N}\) (see (2.2)), the surface area of the \(N-1\)-dimensional unit sphere.Footnote 9 This yields

$$\begin{aligned}&\mathbb {E}\left\{ \#\left\{ \sqrt{N}\varvec{\sigma }\in \mathscr {C}_{N} \left( L\right) :\,g(\varvec{\sigma })\in B' \right\} \right\} = \omega _N\varphi _{\nabla f(\mathbf {n}) }\left( 0\right) \nonumber \\&\quad \times \mathbb {E}\left\{ \left| \det \left( \nabla ^{2}f(\mathbf {n})\right) \right| \mathbf {1}\Big \{\left| \sqrt{N}f(\mathbf {n})-m_{N}\right| <L,\,g \left( \varvec{\sigma }\right) \in B'\Big \}\,\Bigg |\,\nabla f(\mathbf {n})=0\right\} . \end{aligned}$$
(7.45)

In Lemma 13 we stated one case where (7.12) holds as an equality. This case is equivalent to (7.45) without the restrictions on \(g(\varvec{\sigma })\) (after dividing and multiplying by \(\sqrt{Np(p-1)}\) and \((Np(p-1))^\frac{N-1}{2}\) inside the determinant and outside of it, respectively). Thus in order to prove this case what remains is only to check the aforementioned regularity conditions. Those are not difficult to verify when we do not need to worry about the dependence of \(g(\varvec{\sigma })\) and \(f(\varvec{\sigma })\). In particular, the fact that the conditions hold here follows from the case we treat below where we show that they also hold with some \(g(\varvec{\sigma })\) without removing the restrictions on it.

In the other two cases in Lemma 13, (7.12) needs to proved in its original from, with an inequality, for \(2\le i\le 8\); and in a modified form without the condition \(g_1(\varvec{\sigma })\in B_1\) in both sides, for \(i=1\). The proof of the second of the latter two cases is similar to that of the first and therefore we only treat the first. This case is equivalent to (7.12) if \(g(\varvec{\sigma })=(g_i(\varvec{\sigma }),g_1(\varvec{\sigma }))\) and \(B'=B_i^c\times B_1\), and the equality is replaced with an inequality. In order to apply the above to (7.12) directly we will need to verify the regularity conditions and in this case this is not necessarily an easy task. Instead of doing so we relate (7.12) to a similar formula which holds for some modified random fields for which checking the regularity conditions is much easier.

With \(B_{i}^{c}\) denoting the complement of \(B_{i}\) in the corresponding Euclidean space (i.e., in \(\mathbb {R}\) for \(i>2\), \(\mathbb {R}^{\left( N-1\right) ^{2}}\) for \(i=1\), and \(\mathbb {R}^{N-1}\) for \(i=2\)), let \((B_{i}^{c})_\epsilon \) denote the set of points with distance at most \(\epsilon \) from \(B_{i}^{c}\). Let \(Z_{i}\) be a (\(\varvec{\sigma }\)-independent) Gaussian vector in the corresponding Euclidean space all of whose entries are i.i.d standard Gaussian variables. Setting

$$\begin{aligned} h_{i}\left( \varvec{\sigma }\right)&=\left( g_{i}\left( \varvec{\sigma }\right) ,g_{1}\left( \varvec{\sigma }\right) , \sqrt{N}f\left( \varvec{\sigma }\right) \right) ,\\ h_{i,\epsilon }\left( \varvec{\sigma }\right)&=\left( g_{i}\left( \varvec{\sigma }\right) +\epsilon Z_{i},g_{1}\left( \varvec{\sigma }\right) ,\sqrt{N}f \left( \varvec{\sigma }\right) \right) , \end{aligned}$$

we have that, with \(D_{N}=\left( m_{N}-L,\,m_{N}+L\right) \),

$$\begin{aligned}&\mathbb {E}\left\{ \#\left\{ \sqrt{N}\varvec{\sigma }\in \mathscr {C}_{N} \left( L\right) :\,g_{i}\left( \varvec{\sigma }\right) \notin B_{i},\,g_{1} \left( \varvec{\sigma }\right) \in B_{1}\right\} \right\} \\&\quad =\mathbb {E}\left\{ \#\left\{ \varvec{\sigma }\in \mathbb {S}^{N-1}:\nabla f\left( \varvec{\sigma }\right) =0,\,h_{i}\left( \varvec{\sigma }\right) \in B_{i}^{c}\times B_{1}\times D_{N}\right\} \right\} \\&\quad \le \mathbb {E}\left\{ \#\left\{ \varvec{\sigma }\in \mathbb {S}^{N-1}: \nabla f\left( \varvec{\sigma }\right) =0,\,h_{i,\epsilon _{2}}\left( \varvec{\sigma }\right) \in (B_{i}^{c})_{\epsilon _1}\times B_{1}\times D_{N}\right\} \right\} +\delta \left( \epsilon _{1},\epsilon _{2}\right) , \end{aligned}$$

for \(\epsilon _1,\epsilon _2>0\) and an appropriate function \(\delta (\epsilon _1,\epsilon _2)\) satisfying \(\lim _{\epsilon _{2}\rightarrow 0}\delta \left( \epsilon _{1},\epsilon _{2}\right) =0\).

Thus, provided that we can apply (7.45) in the current situation, we have that

$$\begin{aligned}&\mathbb {E}\left\{ \#\left\{ \sqrt{N}\varvec{\sigma }\in \mathscr {C}_{N} \left( L\right) :\,g_{i}\left( \varvec{\sigma }\right) \notin B_{i},\,g_{1} \left( \varvec{\sigma }\right) \in B_{1}\right\} \right\} \\&\quad \le \delta \left( \epsilon _{1},\epsilon _{2}\right) + \omega _{N}\left( Np \left( p-1\right) \right) ^{\frac{N-1}{2}}\varphi _{\nabla f(\mathbf {n})}\left( 0\right) \mathbb {E}\left\{ \left| \det \left( \frac{\nabla ^{2}f(\mathbf {n})}{\sqrt{Np \left( p-1\right) }}\right) \right| \right. \cdots \\&\quad \left. \mathbf {1}\Big \{\left| \sqrt{N}f(\mathbf {n})-m_{N}\right| <L,\,g_{i} \left( \varvec{\sigma }\right) +\epsilon _{2}Z_{i}\in (B_{i}^{c})_{\epsilon _{1}}, \,g_{1}\left( \varvec{\sigma }\right) \in B_{1}\Big \}\,\Bigg |\,\nabla f(\mathbf {n}) =0\right\} . \end{aligned}$$

Since \(g_{i}\left( \varvec{\sigma }\right) +\epsilon _{2}Z_{i}\mathop {\rightarrow }\limits ^{d}g_{i}\left( \varvec{\sigma }\right) \), as \(\epsilon _{2}\rightarrow 0\), and the indicator function of \((B_{i}^{c})_{\epsilon _{1}}\) is upper semi-continuous, the limit of the expectation in the left-hand side above, as \(\epsilon _{2}\rightarrow 0\), is bounded from above by the same expectation with \(\epsilon _{2}=0\). Therefore, by first taking \(\epsilon _{2}\rightarrow 0\) and then taking \(\epsilon _{1}\rightarrow 0\) (and using the monotone convergence theorem and the fact that \(B_i^c\) is closed) the lemma follows.

All that remains is to verify the conditions required for the application of [2, Theorem 12.1.1], i.e., conditions (a)–(g) there, with f, \(\nabla f\) and h of [2, Theorem 12.1.1] being equal to ‘our’ \(\nabla f\left( \varvec{\sigma }\right) \), \(\nabla ^{2}f\left( \varvec{\sigma }\right) \) and \(h_{i,\epsilon }\left( \varvec{\sigma }\right) \). In particular, we need to account for 8 different cases (of i).

From the definition of the Hamiltonian (1.1), the fact that \(f\left( \varvec{\sigma }\right) \) is Gaussian, and from the Borell-TIS inequality [2, Theorem 2.1.1], the components of \(\nabla f\left( \varvec{\sigma }\right) \), \(\nabla ^{2}f\left( \varvec{\sigma }\right) \) and \(h_{i,\epsilon }\left( \varvec{\sigma }\right) \) (in any of the cases above) are continuous and have finite variance at any \(\varvec{\sigma }\).Footnote 10 Combining this with Corollary 18 which assures the non-degeneracy of the Gaussian variable \(\left( f\left( \varvec{\sigma }\right) ,\nabla f\left( \varvec{\sigma }\right) ,\nabla ^{2}f\left( \varvec{\sigma }\right) \right) \), up to symmetry of the Hessian, conditions (a)–(e) can be verified for all the cases. Condition (f) follows since \(\nabla ^{2}f\left( \varvec{\sigma }\right) \) is Gaussian and stationary. Condition (g) which involves the modulus of continuity of the random fields, is verified for \(f\left( \varvec{\sigma }\right) \), \(\nabla f\left( \varvec{\sigma }\right) \) and \(\nabla ^{2}f\left( \varvec{\sigma }\right) \) directly from the definition of the Hamiltonian (1.1).

All that we have left to show is that condition (g) holds for the components of \(g_{i}\left( \varvec{\sigma }\right) +\epsilon Z_{i}\) for any \(1\le i\le 8\) and \(\epsilon >0\). Since \(Z_{i}\) does not depend on \(\varvec{\sigma }\), the components of \(g_{i}\left( \varvec{\sigma }\right) +\epsilon Z_{i}\) and \(g_{i}\left( \varvec{\sigma }\right) \) have the same modulus of continuity, and it is therefore enough to prove that condition (g) holds for the components of \(g_{i}\left( \varvec{\sigma }\right) \). For \(i=1\) this is already proven, since \(g_{1}\left( \varvec{\sigma }\right) \) is equal to \(\nabla ^{2}f\left( \varvec{\sigma }\right) \). For \(i=2,3,4\) (working under the conditioning \(g_{1}\left( \varvec{\sigma }\right) \in B_{1}\)), the modulus of continuity of \(g_{i}\left( \varvec{\sigma }\right) \) can be related to that of the components of \(\left( f\left( \varvec{\sigma }\right) ,\nabla f\left( \varvec{\sigma }\right) ,\nabla ^{2}f\left( \varvec{\sigma }\right) \right) \) to prove condition (g).

In order to deal with the case \(i=5,...,8\), we first note that the modulus of continuity of the supremums in the definition of \(g_{i}\) is bounded by that of the functions the supremum of which is taken. The latter is bounded, up to a constant depending on N, by the sum of moduli of continuity of the components of \(f\left( \varvec{\sigma }\right) \), \(\nabla f\left( \varvec{\sigma }\right) \) and \(\nabla ^{2}f\left( \varvec{\sigma }\right) \). From this condition (g) follows in those cases too and the proof is completed. \(\square \)

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Subag, E., Zeitouni, O. The extremal process of critical points of the pure p-spin spherical spin glass model. Probab. Theory Relat. Fields 168, 773–820 (2017). https://doi.org/10.1007/s00440-016-0724-2

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Keywords

  • Extremal processes
  • Critical points
  • Spin glasses
  • Random fields
  • Random matrices

Mathematics Subject Classification

  • 15A52
  • 82D30
  • 60G60
  • 60G70