On Spin Distributions for Generic p-Spin Models


We provide an alternative formula for spin distributions of generic p-spin glass models. As a main application of this expression, we write spin statistics as solutions of partial differential equations and we show that the generic p-spin models satisfy multiscale Thouless–Anderson–Palmer equations as originally predicted in the work of Mézard–Virasoro (J Phys 46(8):1293–1307, 1985).

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  1. 1.

    Adler, R., Taylor, J.: Random Fields and Geometry. Springer Monographs in Mathematics. Springer, New York (2007)

    Google Scholar 

  2. 2.

    Aldous, D.J.: Exchangeability and Related Topics. Lecture Notes in Mathematics, vol. 1117, pp. 1–198. Springer, Berlin (1985)

    Google Scholar 

  3. 3.

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

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Auffinger, A., Chen, W.-K.: On properties of Parisi measures. Probab. Theory Relat. Fields 161(3–4), 817–850 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Auffinger, A., Chen, W.-K.: The Parisi formula has a unique minimizer. Commun. Math. Phys. 335(3), 1429–1444 (2015)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Auffinger, A., Jagannath, A.: Thouless-Anderson-Palmer equations for the generic p-spin glass model, Ann. Probab., to appear. arXiv:1612.06359

  7. 7.

    Austin, T.: Exchangeable random measures. Ann. Inst. H. Poincaré Probab. Stat. 51(3), 842–861 (2015)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  8. 8.

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Gikhman, I.I., Skorokhod, A.V.: The Theory of Stochastic Processes. I. Classics in Mathematics. Springer, Berlin (2004). Translated from the Russian by S. Kotz, Reprint of the 1974 edition

    Google Scholar 

  10. 10.

    Hoover, D.N.: Row-column exchangeability and a generalized model for probability. In: Exchangeability in Probability and Statistics (Rome, 1981), pp. 281–291. North-Holland, Amsterdam-New York (1982)

  11. 11.

    Jagannath, A., Tobasco, I.: A dynamic programming approach to the Parisi functional. Proc. Am. Math. Soc. 144(7), 3135–3150 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Jagannath, A., Tobasco, I.: Some properties of the phase diagram for mixed p-spin glasses. Probab. Theory Relat. Fields 167, 615–672 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Mézard, M., Parisi, G., Sourlas, N., Toulouse, G., Virasoro, M.: Replica symmetry-breaking and the nature of the spin-glass phase. J. Phys. 45, 843 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Mézard, M., Parisi, G., Virasoro, M.A.: Spin Glass Theory and Beyond, vol. 9. World Scientific, Singapore (1987)

    Google Scholar 

  15. 15.

    Mézard, M., Virasoro, M.A.: The microstructure of ultrametricity. J. Phys. 46(8), 1293–1307 (1985)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Panchenko, D.: The Ghirlanda-Guerra identities for mixed \(p\)-spin model. C. R. Math. Acad. Sci. Paris 348(3–4), 189–192 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Panchenko, D.: The Parisi ultrametricity conjecture. Ann. Math. (2) 177(1), 383–393 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Panchenko, D.: The Sherrington-Kirkpatrick model. Springer, New York (2013)

    Google Scholar 

  19. 19.

    Panchenko, D.: Spin glass models from the point of view of spin distributions. Ann. Probab. 41(3A), 1315–1361 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Stroock, D.W., Srinivasa Varadhan, S.R.: Multidimensional Diffussion Processes, vol. 233. Springer, New York (1979)

    Google Scholar 

  21. 21.

    Talagrand, M.: The Parisi formula. Ann. Math. (2) 163(1), 221–263 (2006)

    MathSciNet  Article  MATH  Google Scholar 

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The authors would like to thank Louis-Pierre Arguin, Gérard Ben Arous, Dmitry Panchenko, and Ian Tobasco for helpful discussions. This research was conducted while A.A. was supported by NSF DMS-1597864 and NSF Grant CAREER DMS-1653552 and A.J. was supported by NSF OISE-1604232.

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Correspondence to Aukosh Jagannath.

A Appendix

A Appendix

A.1 On the Driving Process and Its Descendants

We record here the following basic properties of the driving process, cavity field process, local field process, and magnetization process.

Lemma A.1

Let U be a positive ultrametric subset of a separable Hilbert space that is weakly closed and norm bounded equipped with the restriction of the Borel sigma algebra. Let \(B_{t}(\sigma )\) be the process defined in (1.3). We have the following:

  1. (1)

    The covariance structure is positive semi-definite.

  2. (2)

    There is a version of this process that is jointly measurable and continuous in time.

  3. (3)

    For each \(\sigma ,\)\(B_{t}(\sigma )\) has the law of a brownian motion so that stochastic integration with respect to \(B_{t}(\sigma )\) is well-defined.


We begin with the first. To see this, simply observe that if \(\alpha _{i}\in {\mathbb {R}}\), \((t_{i},\sigma _{i})\) are finitely many points in \([0,q_{*}]\times U\) and \(\sigma _{*}\in U\), then

$$\begin{aligned} \sum \alpha _{i}\alpha _{j}\left( t_{i}\wedge t_{j}\wedge (\sigma _{i},\sigma _{j})\right)&=\sum \alpha _{i}\alpha _{j}\int \mathbb {1}\left\{ {s\le t_{i}}\right\} \mathbb {1}\left\{ {s\le t_{j}}\right\} \mathbb {1}\left\{ {s\le (\sigma _{i},\sigma _{j})}\right\} ds\\&\ge \sum \alpha _{i}\alpha _{j}\int \mathbb {1}\left\{ {s\le t_{i}}\right\} \\&\quad \mathbb {1}\left\{ {s\le t_{j}}\right\} \mathbb {1}\left\{ {s\le (\sigma _{i},\sigma _{*})}\right\} \mathbb {1}\left\{ {s\le (\sigma _{j},\sigma _{*})}\right\} ds\\&=||\sum \alpha _{i}\mathbb {1}\left\{ {s\le t_{i}\wedge (\sigma _{i},\sigma _{*})}\right\} ||_{L^{2}}\ge 0. \end{aligned}$$

We now turn to the second. Observe first that, since \([0,q_{*}]\times U\) is separable and \({\mathbb {R}}\) is locally compact, \(B_{t}(\sigma )\) has a separable version. Furthermore, observe that \(B_{t}(\sigma )\) is stochastically continuous in norm, that is as \((t,\sigma )\rightarrow (t_{0},\sigma _{0})\) in the norm topology, \(P(\left|B_{t}(\sigma )-B_{t_{0}}(\sigma _{0})\right|>\epsilon )\rightarrow 0\). Thus since U is weakly-closed and norm bounded it is compact in the weak topology. Thus it has a version that is jointly measurable by [9, Theorem IV.4.1]. Note then, since the covariance of \(B_{t}(\sigma )\) for fixed \(\sigma \) is that of Brownian motion and \(B_{t}(\sigma )\) is separable, it is in fact continuous by [9, Theorem IV.5.2].

The third property was implicit in the proof of the second. \(\square \)

We now observe the following consequence of the above proposition:

Corollary A.2

Let U be a positive ultrametric subset of a separable Hilbert space that is weakly closed and norm bounded. Then the cavity field process, \(Y_{t}(\sigma )\), the local field process, \(X_{t}(\sigma )\), and the magnetization process, \(M_{t}(\sigma )\), exist, are continuous in time and Borel measurable in \(\sigma .\)

In the above, the following observation regarding the infinitesimal generator of the above processes will be of interest.

Lemma A.3

Let \((\sigma ^{i})_{i=1}^{n}\subset U\) where U is as above. Then we have the following.

  1. (1)

    The driving process satisfies the bracket relation

    $$\begin{aligned} \left\langle B(\sigma ^{1}),B(\sigma ^{2})\right\rangle _{t}={\left\{ \begin{array}{ll} t &{} t\le (\sigma ,\sigma ')\\ 0 &{} t>(\sigma ,\sigma ') \end{array}\right. }. \end{aligned}$$
  2. (2)

    The cavity field process satisfies the bracket relation

    $$\begin{aligned} \left\langle Y(\sigma ^{1}),Y(\sigma ^{2})\right\rangle _{t}={\left\{ \begin{array}{ll} \xi '(t) &{} t\le (\sigma ^{1},\sigma ^{2})\\ 0 &{} else \end{array}\right. }. \end{aligned}$$
  3. (3)

    The local fields process satisfies the bracket relation

    $$\begin{aligned} \left\langle X(\sigma ^{1}),X(\sigma ^{2})\right\rangle _{t}={\left\{ \begin{array}{ll} \xi '(t) &{} t\le (\sigma ^{1},\sigma ^{2})\\ 0 &{} else \end{array}\right. } \end{aligned}$$

    and has infinitesimal generator

    $$\begin{aligned} L_{t}^{lf}=\frac{\xi ''(t)}{2}\left( \sum a_{ij}(t)\partial _{i}\partial _{j}+2\sum b_{i}(t,x)\partial _{i}\right) \end{aligned}$$

    where \(a_{ij}(t)=\mathbb {1}\left\{ {t\le (\sigma ^{i},\sigma ^{j})}\right\} \) and \(b_{i}(t,x)=\zeta ([0,t])\cdot u_{x}(t,x).\)


We begin with the first claim. To see this, observe that by construction,

$$\begin{aligned} B_{t}(\sigma ^{1})=B_{t}(\sigma ^{2}) \end{aligned}$$

for \(t\le (\sigma ^{1},\sigma ^{2})\), thus the bracket above is just the bracket for Brownian motion. If \(t>(\sigma ^{1},\sigma ^{2}):=q\), then the increments \(B_{t}(\sigma ^{1})-B_{q}(\sigma ^{1})\) and \(B_{t}(\sigma ^{2})-B_{q}(\sigma ^{2})\) are independent Brownian motions. This yields the second regime. By elementary properties of Itô processes, we obtain the brackets for \(Y_{t}\) and \(X_{t}\) from this argument. It remains to obtain the infinitesimal generator for the local fields process.

To this end, observe that if \(f=f(t,x_{1},\ldots ,x_{k})\) is a test function, then Itô’s lemma applied to the process \((X_{t}(\sigma ^{i}))_{i=1}^{n}\) yields

$$\begin{aligned} df&=\partial _{t}f\cdot dt+\sum _{i}\partial _{x_{i}}f\cdot dX_{t}(\sigma ^{i})+\frac{1}{2}\cdot \sum \partial _{x_{i}}\partial _{x_{j}}f\cdot d\left\langle X_{t}(\sigma ^{i}),X_{t}(\sigma ^{j})\right\rangle \\&=\left( \partial _{t}f+\sum _{i}\partial _{x_{i}}f\cdot \left( \xi ''(t)\zeta (t)u_{x}(t,X_{t}(\sigma ^{i})\right) \right. \\&\quad \left. +\,\frac{\xi ''}{2}\sum \mathbb {1}\left\{ {t\le (\sigma ^{i},\sigma ^{j})}\right\} \partial _{x_{i}}\partial _{x_{j}}f\right) dt+dMart \end{aligned}$$

where dMart is the increment for some martingale. Taking expectations and limits in the usual fashion then yields the result. \(\square \)

A.2 The Cavity Equations and Ghirlanda–Guerra Identities

In this section, we recall some definitions for completeness. For a textbook presentation, see [18, Chapters 2 and 4]. Let \(\mathscr {M}\) be the set of all measures on the set \(\{-1,1\}^{\mathbb {N}\times \mathbb {N}}\) that are exchangeable, that is, if \((s_i^\ell )\) has law \(\nu \in \mathscr {M}\), then

$$\begin{aligned} (s_{\pi (i)}^{\rho (\ell )}){\mathop {=}\limits ^{(d)}}(s_{i}^{\ell }) \end{aligned}$$

for any permutations \(\pi ,\rho \) of the natural numbers. The Aldous–Hoover theorem [2, 10], states that if \((s_{i}^{\ell })\) is the random variable induced by some measure \(\nu \in \mathscr {M}\), then there is a measurable function of four variables, \(\sigma (w,u,v,x)\), such that

$$\begin{aligned} (s_{i}^{\ell }){\mathop {=}\limits ^{(d)}}(\sigma (w,u_{\ell },v_{i},x_{\ell i})) \end{aligned}$$

where \(w,u_{\ell },v_{i},x_{\ell i}\) are i.i.d. uniform [0, 1] random variables. We call this function a directing function for \(\nu \). The variables \(s_{i}^{\ell }\) are called the spins sampled from \(\nu \).

For any \(\nu \) in \(\mathscr {M}\) with directing function \(\sigma \), let \(\bar{\sigma }(w,u,v)=\int \sigma (w,u,v,x)dx\). Note that since \(\sigma \) is \(\{\pm 1\}\)-valued, this encodes all of the information of \(\sigma (w,u,v,\cdot )\). Define the measure \(\mu \)  on the Hilbert space, \(\mathcal {H}=L^{2}([0,1],dv)\), by the push-forward of du through the map \(u\mapsto \bar{\sigma }(w,u,\cdot )\),

$$\begin{aligned} \mu =(u\mapsto \bar{\sigma }(w,u,\cdot ))_{*}du. \end{aligned}$$

The measure \(\mu \) is called the asymptotic Gibbs measure corresponding to \(\nu \).

A measure \(\nu \) in \(\mathscr {M}\) is said to satisfy the Ghirlanda–Guerra identities if the law of the overlap array satisfies the following property: for every \(f\in C([-1,1]^{n})\) and \(g\in C([-1,1])\), we have

$$\begin{aligned} \mathbb {E}\left\langle f(R^{n})\cdot g(R_{1,n+1})\right\rangle =\frac{1}{n}\left[ \mathbb {E}\left\langle f(R^{n})\right\rangle \cdot \mathbb {E}\left\langle g(R_{12})\right\rangle +\sum _{k=2}^{n}\mathbb {E}\left\langle f(R^{n})\cdot g(R_{1k})\right\rangle \right] , \end{aligned}$$

where by the bracket, \(\left\langle \cdot \right\rangle \), we mean integration against the relevant products of \(\mu \) with itself.

A measure \(\nu \) is said to satisfy the cavity equations if the following is true. Fix the directing function \(\sigma \) and \(\bar{\sigma }\) as above. Let \(g_{\xi '}(\bar{\sigma })\) denote the centered Gaussian process indexed by \(L^2([0,1],dv)\) with covariance

$$\begin{aligned} \mathbb {E}\bigg [g_{\xi '}\big (\bar{\sigma }(w,u,\cdot )\big )g_{\xi '}\big (\bar{\sigma }(w,u',\cdot )\big )\bigg ] = \xi '\bigg (\int \bar{\sigma }(w,u,v)\bar{\sigma }(w,u',v)dv\bigg ) \end{aligned}$$

and let \(G_\xi '(\bar{\sigma })=g_{\xi '}(\bar{\sigma })+z(\xi '(1)-\xi '(||\bar{\sigma }(w,u,\cdot )||_{L^2(dv)}^2))^{1/2}\). Let \(g_{\xi ',i}\) and \(G_{\xi ',i}\) be independent copies of these processes. Let \(n,m,q,r,l\ge 1\) be such that \(n\le m\) and \(l\le q\). Let \(C_l \subset [m]\) and let \(C_l^1=C_l\cap [n]\) and \(C^2_l=C_l\cap (n+[m])\). Let

$$\begin{aligned} U_{l}=\int \mathbb {E}'\prod _{i\in C_{l}^{1}}\tanh G_{\xi ',i}(\bar{\sigma }(w,u,\cdot )\prod _{i\in C_{l}^{2}}\bar{\sigma }_{i}\mathcal {E}_{n,r}du \end{aligned}$$

where \(\mathbb {E}'\) is expectation in z, \(\bar{\sigma }_{i}=\bar{\sigma }(w,u,v_{i}),\theta (t)=\xi '(t)t-\xi (t)\), and where

$$\begin{aligned} \mathcal {E}_{n,r}=\exp \left( \sum _{i\le n}\log \cosh (G_{\xi ',i}(\bar{\sigma }(w,u,\cdot ))+\sum _{k\le r}G_{\theta ,k}(\bar{\sigma }(w,u,\cdot ))\right) . \end{aligned}$$

Let \(V=\mathbb {E}'\mathcal {E}_{n,r}\). The cavity equations for \(n,m,q,r\ge 1\) are then given by

$$\begin{aligned} \mathbb {E}\prod _{l\le q}\mathbb {E}'\prod _{i\in C_{l}}\bar{\sigma }_{i}=\mathbb {E}\frac{\prod _{l\le q}U_{l}}{V^{q}}. \end{aligned}$$

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Auffinger, A., Jagannath, A. On Spin Distributions for Generic p-Spin Models. J Stat Phys 174, 316–332 (2019). https://doi.org/10.1007/s10955-018-2188-5

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  • Spin glasses
  • Thouless–Anderson–Palmer equations
  • Cavity equations