arXiv subjects

Ali Talebi

Publications and source records attributed to Ali Talebi.

6 recordsLinked to original sources

arXiv2025

Equations of state and stability condition of mixed p-spin glass model

The Sherrington-Kirkpatrick (SK) is a foundational model for understanding spin glass systems. It is based on the pairwise interaction between each two spins in a fully connected lattice with quenched disordered interactions. The nature of long-range interaction among spins in the (SK) model simplifies the study of this system by eliminating fluctuations. An advanced (SK) model, known as the p-spin model, introduces higher-order interactions that involve the interaction of P spins. This research focuses on the general Hamiltonian of the spin glass model with long-range interaction, referred to as the mixed p-spin glass model, which consists of adding all p-spin interaction terms. This research aims to derive the equation of states for this Hamiltonian, formulate the equation of state within the framework of the first replica symmetry breaking, and determine both the stability condition of the replica symmetric solution and the stability of the replicas belonging to the same group in the first step of replica symmetry breaking.

cond-mat.dis-nn↗

arXiv2024

Single replica spin-glass phase detection using field variation and machine learning

The Sherrington-Kirkpatrick spin-glass model used the replica symmetry method to find the phase transition of the system. In 1979-1980, Parisi proposed a solution based on replica symmetry breaking (RSB), which allowed him to identify the underlying phases of complex systems such as spin-glasses. Regardless of the method used for detection, the intrinsic phase of a system exists whether or not replicas are considered. We introduce a single replica method of spin-glass phase detection using the field's variation experienced by each spin in a system configuration. This method focuses on a single replica with quenched random couplings. Each spin inevitably observes a different field from the others. Our results show that the mean and variance of fields named "Spontaneous Configurational Field" experienced by spins are suitable indicators to explore different ferromagnetic, paramagnetic, and mixed phases. To classify different phases of the system with defined indicators we have developed an algorithm based on machine learning to analyze the desired samples.

cond-mat.dis-nn↗

arXiv2018

Non-commutative Stein inequality and its applications

The non-commutative Stein inequality asks whether there exists a constant $C_{p,q}$ depending only on $p, q$ such that \begin{equation*} \left\| \left(\sum_{n} |\mathcal{E}_{n} (x_n) |^{q}\right)^{\frac{1}{q}} \right\|_p \leq C_{p,q} \left\| \left(\sum_{n} | x_n |^q \right)^{\frac{1}{q}}\right \|_p\qquad \qquad (S_{p,q}), \end{equation*} for (positive) sequences $(x_n)$ in $L_p(\mathcal{M})$. The validity of $(S_{p,2})$ for $1 < p < \infty$ and $(S_{p,1})$ for $1 \leq p < \infty$ are known. In this paper, we verify (i) $(S_{p,\infty})$ for $1 < p \leq \infty$; (ii) $(S_{p,p})$ for $1 \leq p < \infty$; (iii) $(S_{p,q})$ for $1 \leq q \leq 2$ and $q<p<\infty$. We also present some applications.

math.OA↗

arXiv2018

A variance bound for a general function of independent noncommutative random variables

The main purpose of this paper is to establish a noncommutative analogue of the Efron--Stein inequality, which bounds the variance of a general function of some independent random variables. Moreover, we state an operator version including random matrices, which extends a result of D. Paulin et al. [Ann. Probab. 44 (2016), no. 5, 3431--3473]. Further, we state a Steele type inequality in the framework of noncommutative probability spaces.

math.FA↗

arXiv2017

Etemadi and Kolmogorov inequalities in noncommutative probability spaces

Based on a maximal inequality type result of Cuculescu, we establish some noncommutative maximal inequalities such as Haj\'ek--Penyi inequality and Etemadi inequality. In addition, we present a noncommutative Kolmogorov type inequality by showing that if $x_1, x_2, \ldots, x_n$ are successively independent self-adjoint random variables in a noncommutative probability space $(\mathfrak{M}, \tau)$ such that $\tau\left(x_k\right) = 0$ and $s_k s_{k-1} = s_{k-1} s_k$, where $s_k = \sum_{j=1}^k x_j$, then for any $\lambda > 0$ there exists a projection $e$ such that $$1 - \frac{(\lambda + \max_{1 \leq k \leq n} \|x_k\|)^2}{\sum_{k=1}^n {\rm var}(x_k)}\leq \tau(e)\leq \frac{\tau(s_n^2)}{\lambda^2}.$$ As a result, we investigate the relation between convergence of a series of independent random variables and the corresponding series of their variances.

math.OA↗

arXiv2017

Noncommutative Blackwell-Ross martingale inequality

We establish a noncommutative Blackwell--Ross inequality for supermartingales under a suitable condition which generalize Khan's works to the noncommutative setting. We then employ it to deduce an Azuma-type inequality.

math.OA↗