Abstract
We provide an alternative formula for spin distributions of generic pspin 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 pspin 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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Acknowledgements
The authors would like to thank LouisPierre Arguin, Gérard Ben Arous, Dmitry Panchenko, and Ian Tobasco for helpful discussions. This research was conducted while A.A. was supported by NSF DMS1597864 and NSF Grant CAREER DMS1653552 and A.J. was supported by NSF OISE1604232.
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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)
The covariance structure is positive semidefinite.

(2)
There is a version of this process that is jointly measurable and continuous in time.

(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 welldefined.
Proof
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
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, \(P(\leftB_{t}(\sigma )B_{t_{0}}(\sigma _{0})\right>\epsilon )\rightarrow 0\). Thus since U is weaklyclosed 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)
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)
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)
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
$$\begin{aligned} L_{t}^{lf}=\frac{\xi ''(t)}{2}\left( \sum a_{ij}(t)\partial _{i}\partial _{j}+2\sum b_{i}(t,x)\partial _{i}\right) \end{aligned}$$(A.1)where \(a_{ij}(t)=\mathbb {1}\left\{ {t\le (\sigma ^{i},\sigma ^{j})}\right\} \) and \(b_{i}(t,x)=\zeta ([0,t])\cdot u_{x}(t,x).\)
Proof
We begin with the first claim. To see this, observe that by construction,
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
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
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
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 pushforward of du through the map \(u\mapsto \bar{\sigma }(w,u,\cdot )\),
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
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
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
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
Let \(V=\mathbb {E}'\mathcal {E}_{n,r}\). The cavity equations for \(n,m,q,r\ge 1\) are then given by
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Auffinger, A., Jagannath, A. On Spin Distributions for Generic pSpin Models. J Stat Phys 174, 316–332 (2019). https://doi.org/10.1007/s1095501821885
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Keywords
 Spin glasses
 Thouless–Anderson–Palmer equations
 Cavity equations