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The consistency of a physical interpretation of Parisi's (1979, 1980) replica-symmetry-breaking ansatz using the scheme of pure states is examined by means of numerical solutions of the Thouless-Anderson-Palmer (TAP) equations (1977). A good agreement between the result from the TAP theory and from Parisi's replica solution is obtained for the order parameter function and the internal field distribution. To this extent, the static interpretation of Parisi's solution is correct for the Sherrington-Kirkpatrick spin glass model. As a by-product of the evaluation of Parisi's solution it is found that the Parisi-Toulouse hypothesis holds well numerically. Furthermore, a new microscopic projection hypothesis is proposed.

A simple algebraic derivation of Parisi's equations (1981) for the Sherrington-Kirkpatrick model for spin glasses is given.

In this work, extensive numerical simulations and scaling analysis are performed to investigate competitive growth between the linear and nonlinear stochastic dynamic growth systems, which belong to the Edwards-Wilkinson (EW) and Kardar-Parisi-Zhang (KPZ) universality classes, respectively. The linear growth systems include the EW equation and the random deposition with surface relaxation (RDSR) model, the nonlinear growth systems involve the KPZ equation and typical discrete models including ballistic deposition (BD), Etching, and restricted solid-on-solid (RSOS). The scaling exponents are obtained in both the (1+1)- and (2+1)-dimensional competitive growth with the nonlinear growth probability $p$ and the linear proportion $1-p$. Our results show that, when $p$ changes from 0 to 1, there exist non-trivial crossover effects from EW to KPZ universality classes based on different competitive growth rules. Furthermore, the growth rate and the porosity are also estimated within various linear and nonlinear growth of cooperation and competition.

The theoretical physicist Giorgio Parisi from the University of Roma I "La Sapienza" has been awarded the 2021 Wolf Prize

The Nobel Laureate Giorgio Parisi is interviewed by JPhys Complexity Editor-in-Chief, Ginestra Bianconi, on themes related to the 2021 Nobel Prize in Physics awarded to him for research on complex systems.

The stability of Parisi's solution (1979-80) of the infinite-range spin glass model is examined near the transition temperature. It is found that the quadratic terms in the variation of the free energy are all non-positive. The free energy is only a local maximum with respect to q(x) if q is constrained to be a monotonic function of x. The connection between replica symmetry breaking and failure of linear response theory for magnetic susceptibility is shown in more detail. It is shown that the order parameter susceptibility is divergent, as expected, in the spin glass phase.

The m-component SK model of a spin glass is studied in the presence of a magnetic field H. Below the Gabay-Toulouse line, corresponding to the freezing of the transverse components, the replica symmetry is broken a la Parisi. Near T_c=1 analytic expressions for the free energy, the longitudinal (q₁) and transverse (q_T) order parameters are derived. Different behaviours are found in 'strong' (H²>>(1-T)³) and 'weak' (H²<or approximately=(1-T)³) fields, with a likely crossover. A brief discussion of stability is given. Possible relevance to experiments is examined.

The authors argue that standard treatments of dynamics are incorrect for non-ergodic systems such as the Sherrington-Kirkpatrick model of a spin glass. Since the system never loses the memory of its initial state it is necessary to average over initial states with a Boltzmann weight to obtain time-dependent correlation functions as conventionally defined. It is now necessary to introduce replicas. The 'dynamical' order parameter, q_EA, is equal to q(x=1) in Parisi's scheme whereas the statistical mechanics order parameter, q, is given by integral ₀^Iq(x) dx. They also propose an interpretation of the function q(x).

The concept of a state overlap function, suitable for computer simulation and relatable to the Parisi order function of mean-field theory, is extended to vector spin glasses. Application is made to the Parisi predictions for continuous and discrete vector spins, as germane respectively to Heisenberg and Potts systems.

Generating functions of the form Z(u)= Sigma _s exp(u beta F_s) are considered, where the sum is over solutions of the Thouless-Anderson-Palmer (TAP) equations for the Sherrington-Kirkpatrick spin glass model and F_s is the free energy of solution s. It is shown that the weight function exp(u beta F_s)/Z(u) projects out solutions of the lowest free energy for all u less than a critical value. The associated Parisi order parameter function q(x) has the scaling form q(x)=q_P( mod u mod x) for mod u mod x<x, where q_p(x) is the Parisi function for the canonical weight (u=-1) and x is the 'break point' in the Parisi function.