Related papers: Exponential control of overlap in the replica meth…
We investigate the large deviation behavior of the overlap probability density in the Sherrington--Kirkpatrick model from several analytical perspectives. First we analyze the spin glass phase using the coupled replica scheme. Here…
In order to study certain questions concerning the distribution of the overlap in Sherrington--Kirkpatrick type models, such as the chaos and ultrametricity problems, it seems natural to study the free energy of multiple systems with…
The validity of the Parisi formula in the Sherrington-Kirkpatrick model (SK) was initially proved by Talagrand [18]. The central argument therein relied on a very dedicated study of the coupled free energy via the two-dimensional…
We compute the probability of positive large deviations of the free energy per spin in mean-field Spin-Glass models. The probability vanishes in the thermodynamic limit as $P(\Delta f) \propto \exp[-N^2 L_2(\Delta f)]$. For the…
Spin glass models involving multiple replicas with constrained overlaps have been studied in [FPV92; PT07; Pan18a]. For the spherical versions of these models [Ko19; Ko20] showed that the limiting free energy is given by a Parisi type…
By controlling quantum fluctuations via the Falk-Bruch inequality we give the first rigorous argument for the existence of a spin-glass phase in the quantum Sherrington-Kirkpatrick model with a transverse magnetic field if the temperature…
In a region above the Almeida-Thouless line, where we are able to control the thermodynamic limit of the Sherrington-Kirkpatrick model and to prove replica symmetry, we show that the fluctuations of the overlaps and of the free energy are…
Based on \cite{H}, it is well known that the rescaled two point correlation functions \[ \sqrt{N} \langle \sigma_i ; \sigma_j\rangle = \sqrt{N} \big( \langle \sigma_i \sigma_j\rangle -\langle \sigma_i\rangle \langle \sigma_j\rangle\big) \]…
We consider the random fluctuations of the free energy in the $p$-spin version of the Sherrington-Kirkpatrick model in the high temperature regime. Using the martingale approach of Comets and Neveu as used in the standard SK model combined…
We study the Replica Symmetric region of general multi-species Sherrington-Kirkpatrick (MSK) Model and answer some of the questions raised in Ann.~Probab.~43~(2015), no.~6, 3494--3513, where the author proved the Parisi formula under…
We prove a central limit theorem for the normalized overlap between two replicas in the spherical SK model in the high temperature phase. The convergence holds almost surely with respect to the disorder variables, and the inverse…
In this paper we obtain two results for the Sherrington-Kirkpatrick (SK) model, and we show that they both emerge from a single approach. First, we prove that the average of the overlap takes positive values when it is non zero. More…
A quantum Parisi formula for the transverse field Sherrington-Kirkpatrick (SK) model is proven with an elementary mathematical method. First, a self-overlap corrected quantum model of the transverse field SK model is represented in terms of…
In this note, the Sherrington Kirkpatrick model of interacting spins is under consideration. In the high temperature region, we give an asymptotic expansion for the expected value of some genereral polynomial of the overlap of the system…
We consider the Sherrington--Kirkpatrick spin glass model with zero external field and at inverse temperature $\beta>0$. Let $F_N(\beta)$ be the corresponding log-partition function. Under the assumption that $c_N:=N^{1/3}(1-\beta_N^2)$ is…
In this note, we consider a SK (Sherrington--Kirkpatrick)-type model on Z^d for d greater or equal to 1, weighted by a function allowing to any single spin to interact with a small proportion of the other ones. In the thermodynamical limit,…
We argue that when the number of spins $N$ in the SK model is finite, the Parisi scheme can be terminated after $K$ replica-symmetry breaking steps, where $K(N) \propto N^{1/6}$. We have checked this idea by Monte Carlo simulations: we…
We calculate the probability distribution of the overlap between a spin glass and a copy of itself in which the bonds are randomly perturbed in varying degrees. The overlap distribution is shown to go to a delta distribution in the…
The Sherrington-Kirkpatrick spin-glass model is investigated by means of Monte Carlo simulations employing a combination of the multi-overlap algorithm with parallel tempering methods. We investigate the finite-size scaling behaviour of the…
We prove a simple identity relating the $k$th moment of the partition function $Z_N(\cdot)$ in the SK model to the $N$th moment of the partition function $Z_k(\cdot)$. As a corollary we find a characterisation of the limit $\lim_{N \to…