The Full Replica Symmetry Breaking in the Ising Spin Glass on Random Regular Graph

Abstract

In this paper, we extend the full replica symmetry breaking scheme to the Ising spin glass on a random regular graph. We propose a new martingale approach, that overcomes the limits of the Parisi–Mézard cavity method, providing a well-defined formulation of the full replica symmetry breaking problem in random regular graphs. Finally, we define the order parameters of the system and get a set of self-consistency equations for the order parameters and the free energy. We face up the problem only from a technical point of view: the physical meaning of this approach and the quantitative evaluation of the solution of the self-consistency equations will be discussed in next works.

Appendix: The 1 RSB Free Energy Functional

In this appendix, we show that the replica symmetry breaking ansatz described in the Sect. 3.1 reproduces the

Let us define the functions

$$\begin{aligned} F^{\text {(v)}}(J_{0,1},\ldots ,J_{0,c},h_1, \ldots ,h_c\,)=\log \Delta ^{\text {(v)}}(J_{0,1}, \ldots ,J_{0,c},h_1,\ldots ,h_c), \end{aligned}$$

(77)

and

$$\begin{aligned} F^{\text {(e)}}(J_{1,2},h_1,h_2\,)= \Delta ^{\text {(e)}}(J_{1,2},h_{1},h_{2}). \end{aligned}$$

(78)

Putting $r=1$, the cavity field functional (16) is a measurable function of two independent normal random variables $W^{(0)}$ and $W^{(1)}$:

$$\begin{aligned} \{W^{(0)},W^{(1)}\}\mapsto h\big (\,W^{(0)},W^{(1)}\,\big ). \end{aligned}$$

(79)

The edge and vertex contributions to the free energy are given by a single iteration of the iterative rule (21):

$$\begin{aligned}&\phi ^{\text {(e/v)}}_{0}\big (\varvec{J},\,\varvec{W}^{(0)}\,\big )=\frac{1}{x_{1}}\log \mathbb {E}_{\mathbb {W}_1^{\otimes (2/c)}}\left[ \,\exp \left( \,x_{1} \phi ^{\text {(e/v)}}_{1}\big (\varvec{J},\varvec{W}^{(0)},\, \varvec{W}^{(1)}\,\big ) \,\right) \,\Big |\varvec{W}^{(0)}\right] \nonumber \\&\quad =\frac{1}{x_{1}}\log \left( \int \left( \prod ^{(2/c)}_{i=1} d\nu \big (W^{(1)}_{i}\big )\,\right) e^{\,x_{1}\, F^{\text {(e/v)}}\left( \varvec{J},\varvec{h}\big (\,\varvec{W}^{(0)},\,\varvec{W}^{(1)}\,\big )\,\,\right) \,}\,\,\right) , \end{aligned}$$

(80)

where $\nu$ is the normal distribution.

The free energy functional is finally given by

$$\begin{aligned} \Phi =\overline{\int \prod ^c_{i=1} d\nu \big (W_i^{(0)}\,\big )\,\,\frac{1}{x_{1}} \log \left( \int \left( \prod ^{c}_{i=1} d\nu \big (W^{(1)}_{i}\big )\,\right) e^{\,x_{1}\, F^{\text {(v)}}\left( J_{0,1},\ldots ,J_{0,c},\,h_ 1\big (\,W_1^{(0)},\,W_1^{(1)}\,\big ), \ldots \,h_c\big (\,W_c^{(0)}, \,W_c^{(1)}\,\big )\,\right) \,}\,\,\right) }\nonumber \\ -\frac{c}{2}\overline{\int d\nu \big (W_1^{(0)}\big )d\nu \big (W_2^{(0)}\big ) \frac{1}{x_{1}}\log \left( \int _{\mathbb {R}^{(2/c)}}d \nu \big (W_1^{(1)}\big )d\nu \big (W_2^{(1)}\big )\,e^{\,x_{1}\, F^{\text {(e)}}\left( J_{1,2},h_1\big (\,W_1^{(0)}, \,W_1^{(1)}\,\big ),\,h_2\big (\,W_2^{(0)}, \,W_2^{(1)}\,\big )\,\right) \,}\,\,\right) }.\nonumber \\ \end{aligned}$$

(81)

Now, let us define the probability density distribution of the cavity field h, conditionally to a fixed value for the random variable $W^{(0)}$:

$$\begin{aligned}&\widehat{\pi }\big (y\big |W^{(0)}\,\big )=\mathbb {E}_{\mathbb {W}_1^{\otimes (2/c)}}\left[ \delta \big (\,y-h\big (\,W^{(0)},W^{(1)}\,\big )\,\big )\Big |W^{(0)}\right] \nonumber \\&\quad =\int d\nu \big (W^{(1)}\big )\delta \big (\,h\big (\,W^{(0)},W^{(1)}\,\big )-y\,\big ),\quad y\in \mathbb {R}, \end{aligned}$$

(82)

where the symbol $\delta (\cdot )$ denotes the Dirac delta distribution. For each fixed value of $y\in \mathbb {R}$, the quantity $\widehat{\pi }(y|\,\cdot \,)$ is a positive random variable, since it depends on $W^{(0)}$. Then, we can define the probability density distribution of the random probability density distribtion $\widehat{\pi }:=\big \{\widehat{\pi }\big (y\big |\,\cdot \,\big );\,y\in \mathbb {R}\,\big \}$ in such a way:

$$\begin{aligned} \Pi [\,\pi \,]=\mathbb {E}_{\mathbb {W}_0^{\otimes (2/c)}}\left[ \delta \big [\,\pi \big (\,\cdot \,\big )-\widehat{\pi }\big (\,\cdot \,\big |W^{(0)}\,\big )\,\big ]\,\right] =\int d\nu \big (W^{(0)}\big )\delta \big [\,\pi \big (\,\cdot \,\big )-\widehat{\pi }\big (\,\cdot \,\big |W^{(0)}\,\big )\,\big ], \end{aligned}$$

(83)

where $\delta [\,\cdot \,]$ is the functional Dirac delta. Substituting the Eqs. (82) and (83) in(81), one get

$$\begin{aligned}&x_1 \Phi =x_1 \Phi \big [\Pi ,\pi ,x_1\big ]=\overline{\int \left( \prod ^c_{i=1} d[\pi _i]\Pi [\,\pi _i\,]\right) \,\,\log \left( \int \left( \prod ^{c}_{i=1} dy \,\pi \big (y\big )\,\right) e^{\,x_{1}\, F^{\text {(v)}}\left( J_{0,1},\ldots ,J_{0,c}, \,y_1,\ldots \,y_c\,\big )\,\right) }\right) }\nonumber \\&\quad -\frac{c}{2}\overline{\int d[\pi _1]\Pi [\,\pi _1\,]\,\,d[\pi _2]\Pi [\,\pi _2\,] \log \left( \int _{\mathbb {R}^{(2/c)}}\,dy_1\,\pi _i(y_1)\,dy_2\,\pi _i(y_2)\,e^{\,x_{1}\, F^{\text {(e)}}\left( J_{1,2},y_1,\,y_2\right) }\right) }, \end{aligned}$$

(84)

where the symbol $\int d[\pi ]$ represents the functional integral.

The variational free energy functional (84) is equivalent to a finite temperature version of the 1-RSB variational free energy presented in Appendix B of [27]. The functional $\Pi$ is the 1-RSB cavity method order parameter and $x_1$ is the Parisi 1-RSB parameter.