arXiv subjects

Chokri Manai

Publications and source records attributed to Chokri Manai.

At least 19 recordsLinked to original sources

arXiv2026

Correlation inequalities for transversal field models with application to quantum glasses

We study the class of transversal field models with pair interactions, including the quantum Curie-Weiss and the transversal field Ising model, as well as quantum spin glasses based on the Sherrington-Kirkpatrick (SK) and Edwards-Anderson models. Our main result is a general correlation inequality based on a Gaussian convolution estimate, which is derived from an extension of the Ding-Song-Sun inequality and an application of Brascamp-Lieb convexity techniques. The results apply for sufficiently strong transversal fields and, for quantum spin glasses, yield a high-field regime without replica order. Combined with the previously established low-field replica-symmetry-breaking regime, this gives distinct low- and high-field overlap phases. As a corollary of the logarithmic Sobolev inequalities for the underlying functional integrals derived here, we show the concentration of the self- and replica-overlap, and determine the high-field asymptotics of the pressure in the quantum SK model.

math-ph↗

arXiv2026

Digit Mixing under Polynomial Maps

Let $X=\sum_{n\geq1}\xi_n2^{-n} $ be a random number where we model the digits $\xi_n$ as independent Bernoulli random variables with possibly non-identical parameters $p_n=\mathbb{P}(\xi_n=1)$. For any polynomial $P\in\mathbb{R}[X]$ with degree $d\geq2$, we prove almost sure absolute normality of $P(X)$ under the condition $p_n(1-p_n)\geq (\log n)^{\Gamma(d)} n^{-(d-1)/d}$ for a suitable constant $\Gamma(d)$ depending only on the degree $d$. Our analysis reveals the sharp power law $n^{-(d-1)/d}$, which is suggested by an elementary heuristics regarding carrier interactions. Our results establish a transition as we further show that the pure critical power law is insufficient, but the precise critical window remains an interesting open problem. As far as we know, this is the first sharp result on digit mixing. We complement our main results by structurally convenient summability criteria, which turns out to be sharp at least for $X^2$, and we formulate a more general conjecture for higher degrees. Our proofs rely on Fourier decay estimates which we obtain by probabilistic argument involving conditioning and non-resonancy estimates combined with a subtle triangularization argument.

math.PR↗

arXiv2026

The quantum Almeida-Thouless line in the self-overlap-corrected quantum Sherrington-Kirkpatrick model

We present a complete analysis of the glass transition in the self-overlap-corrected Sherrington-Kirkpatrick (SK) model in a transverse magnetic field, also referred to as the quantum SK model. In particular, we determine the phase boundary separating the glassy and paramagnetic phases. The proof is based on a simplified Parisi variational principle for the quantum pressure, which only involves classical Parisi order parameters. As part of the proof, we also analyze the pressure of the self-overlap-constrained quantum SK model and its Parisi description, as well as the pressure of generalized quantum Hopfield models.

math-ph↗

arXiv2025

Quantum Mean-Fields Spin Systems in a Random External Field

In this work, we consider general exchangeable quantum mean-field Hamiltonian such as the prominent quantum Curie-Weiss model under the influence of a random external field. Despite being arguably the simplest class of disordered quantum systems, the random external field breaks the symmetry of the mean-field Hamiltonian and hence standard quantum de Finetti type or semiclassical arguments are not directly applicable. We introduce a novel strategy in this context, which can be seen as non-commutative large deviation analysis, allowing us to characterize the limiting free energy in terms of a simple and explicit variational formula. The proposed method is general enough to be used for other classes of mean-field models such as multi species Hamiltonians.

math-ph↗

arXiv2025

Transcendence Meets Normality: Construction of Transcendentally Normal Numbers

In this work, we study real numbers $x$ for which $p(x)$ is (absolutely) normal for every non-constant integer-valued polynomial $p$. We call such numbers transcendentally normal. We prove that almost every real number is transcendentally normal and provide an explicit construction of such a number, based on Sierpinski's covering method and novel ideas involving the so-called stretch function. In the next step, we transform this construction into an algorithm that computes the digits of a t-normal number recursively in all integer bases. Moreover, we extend our covering approach to construct and compute LIL-normal numbers whose discrepancies are of the order predicted by the law of the iterated logarithm. We also take the opportunity to discuss several interesting open problems.

math.NT↗

arXiv2025

Dynamical Phase Diagram of the REM under independent spin-flips

We study the energy landscape of the Random Energy model (REM) integrated along trajectories of the simple random walk on the hypercube. We show that the quenched cumulant generating function of the time integral of the REM energy undergoes phase transitions in the large $N$ limit for trajectories of any time extent, and identify phases distinguished by the activity and value of the time integral. This is achieved by relating the dynamical behavior to the spectral properties of Hamiltonians associated with the Quantum Random Energy Model (QREM). Of independent interest are deterministic $ \ell^p $-properties of the resolvents of such Hamiltonians, which we establish.

math-ph↗

arXiv2025

Super-Dense Sets and Their Role in the Theory of Normal Numbers

We introduce and study a new topological notion of the size for subsets of the real line, called \emph{super-density}. A set $A\subset\mathbb{R}$ is super-dense if for every non-empty open interval $I$ and every nowhere constant continuous function $\varphi\colon I\to\mathbb{R}$, we have $\varphi(I\cap A)\cap A\neq\emptyset$. We first establish basic properties of super-dense sets. Our main topological result characterizes them within the framework of Baire category: a set with the Baire property is super-dense if and only if it is co-meager. We then investigate the implications for the theory of normal numbers. We prove that the set of non-normal numbers is super-dense, whereas the set of normal numbers is not. Consequently, no nowhere constant continuous function can map all non-normal numbers to normal numbers. Conversely, we explicitly construct a computable nowhere constant continuous function that maps all normal numbers to non-normal numbers. Finally, we provide a constructive algorithm that, given any countable family of nowhere constant continuous functions, produces a real number $x$ such that $x$ and all its images under these functions are non-normal. As a corollary, we obtain the existence of a non-normal number $x$ such that $e^{\alpha x}$ is non-normal for every non-zero algebraic $\alpha$.

math.NT↗

arXiv2025

The Quantum Random Energy Model is the Limit of Quantum $ p $-Spin Glasses

We consider the free energy of a class of spin glass models with $ p$-spin interactions in a transverse magnetic field. As $ p \to \infty $, the infinite system-size free energy is proven to converge to that of the quantum random energy model. This is accomplished by combining existing analytical techniques addressing the non-commutative properties of such quantum glasses, with the description of the typical geometry of extreme negative deviations of the classical $ p $-spin glass. We also review properties of the corresponding classical free energy and conjectures addressing $ 1/p $-corrections in the quantum case.

math-ph↗

arXiv2024

A Parisi Formula for Quantum Spin Glasses

We establish three equivalent versions of a Parisi formula for the free energy of mean-field spin glasses in a transversal magnetic field. These results are derived from available results for classical vector spin glasses by an approximation method using the functional integral representation of the partition function. In this approach, the order parameter is a non-decreasing function with values in the non-negative, real hermitian Hilbert-Schmidt operators. For the quantum Sherrington-Kirkpatrick model, we also show that under the assumption of self-averaging of the self-overlap, the optimising Parisi order parameter is found within a two-dimensional subspace spanned by the self-overlap and the fully stationary overlap.

cond-mat.dis-nn↗

arXiv2023

Transcendental properties of entropy-constrained sets: Part II

In this work, we address the question of the impossibility of certain single-letter formulas by exploiting the semi-algebraic nature of various entropy-constrained sets. The focus lies on studying the properties of the level sets of relative entropy, mutual information, and R\'{e}nyi entropies. We analyze the transcendental structure of the set of states in which one of the aforementioned entropy quantities is fixed. Our results rule out (semi)algebraic single-shot characterizations of these entropy measures with bounded ancilla for both the classical and quantum cases.

math-ph↗

arXiv2023

The Spectral Gap and Low-Energy Spectrum in Mean-Field Quantum Spin Systems

A semiclassical analysis based on spin-coherent states is used to establish a classification and formulae for the spectral gap of mean-field spin Hamiltonians. For gapped systems we provide a full description of the low-energy spectra based on a second-order approximation to the semiclassical Hamiltonian hence justifying fluctuation theory at zero temperature for this case. We also point out a shift caused by the spherical geometry in these second-order approximations.

math-ph↗

arXiv2022

Trajectory phase transitions in non-interacting systems: all-to-all dynamics and the random energy model

We study the fluctuations of time-additive random observables in the stochastic dynamics of a system of $N$ non-interacting Ising spins. We mainly consider the case of all-to-all dynamics where transitions are possible between any two spin configurations with uniform rates. We show that the cumulant generating function of the time-integral of a normally distributed quenched random function of configurations, i.e., the energy function of the random energy model (REM), has a phase transition in the large $N$ limit for trajectories of any time extent. We prove this by determining the exact limit of the scaled cumulant generating function. This is accomplished by connecting the dynamical problem to a spectral analysis of the all-to-all quantum REM. We also discuss finite $N$ corrections as observed in numerical simulations.

cond-mat.stat-mech↗

arXiv2022

Spectral Analysis of the Quantum Random Energy Model

The Quantum Random Energy Model (QREM) is a random matrix of Anderson-type which describes effects of a transversal magnetic field on Derrida's spin glass. The model exhibits a glass phase as well as a classical and a quantum paramagnetic phase. We analyze in detail the low-energy spectrum and establish a localization-delocalization transition for the corresponding eigenvectors of the QREM. Based on a combination of random matrix and operator techniques as well as insights in the random geometry, we derive next-to-leading order asymptotics for the ground-state energy and eigenvectors in all regimes of the parameter space. Based on this, we also deduce the next-to-leading order of the free energy, which turns out to be deterministic and on order one in the system size in all phases of the QREM. As a result, we determine the nature of the fluctuations of the free energy in the spin glass regime.

math-ph↗

arXiv2021

Existence of replica-symmetry breaking in quantum glasses

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 and the field are sufficiently low. The argument also applies to the generalization of the model with multi-spin interactions, sometimes dubbed as transverse $p$-spin model.

cond-mat.dis-nn↗

arXiv2021

The de Almeida-Thouless Line in Hierarchical Quantum Spin Glasses

We determine explicitly and discuss in detail the effects of the joint presence of a longitudinal and a transversal (random) magnetic field on the phases of the Random Energy Model (REM) and its hierarchical generalization, the GREM. Our results extent known results both in the classical case of vanishing transversal field and in the quantum case for vanishing longitudinal field. Following Derrida and Gardner, we argue that the longitudinal field has to be implemented hierarchically also in the Quantum GREM. We show that this ensures the shrinking of the spin glass phase in the presence of the magnetic fields as also expected for the Quantum Sherrington-Kirkpatrick model.

math-ph↗

arXiv2020

Generalized Random Energy Models in a Transversal Magnetic Field: Free Energy and Phase Diagrams

We determine explicit variational expressions for the free energy of mean-field spin glasses in a transversal magnetic field, whose glass interaction is given by a hierarchical Gaussian potential as in Derrida's Generalized Random Energy Model (GREM), its continuous version (CREM) or the non-hierarchical GREM. The corresponding phase diagrams, which generally include glass transitions as well as magnetic transitions, are discussed. In the glass phase, the free energy is generally determined by both the parameters of the classical model and the transversal field.

math-ph↗

arXiv2019

The Quantum Random Energy Model as a Limit of p-Spin Interactions

We consider the free energy of a mean-field quantum spin glass described by a $ p $-spin interaction and a transversal magnetic field. Recent rigorous results for the case $ p= \infty $, i.e. the quantum random energy model (QREM), are reviewed. We show that the free energy of the $ p $-spin model converges in a joint thermodynamic and $ p \to \infty $ limit to the free energy of the QREM.

math-ph↗

arXiv2019

Phase Diagram of the Quantum Random Energy Model

We prove Goldschmidt's formula [Phys. Rev. B 47 (1990) 4858] for the free energy of the quantum random energy model. In particular, we verify the location of the first order and the freezing transition in the phase diagram. The proof is based on a combination of variational methods on the one hand, and percolation bounds on large-deviation configurations in combination with simple spectral bounds on the hypercube's adjacency matrix on the other hand.

math-ph↗