arXiv subjects

Isaac Pérez Castillo

Publications and source records attributed to Isaac Pérez Castillo.

At least 19 recordsLinked to original sources

arXiv2026

Spectral Signatures of Replica Symmetry Breaking in Optimization-Induced Random Matrices

We study optimization-induced matrix ensembles generated by Gibbs measures. The same quenched disorder that weights configurations also supplies the matrix entries observed on them. For glassy Gibbs measures this raises a natural question: does the induced spectrum inherit the underlying glassy Gibbs geometry? In a dense tensor optimization model we find a selective answer. A single induced matrix has a universal leading bulk that washes out the glassy organization. The difference of two matrices built from independent thermal samples in the same disorder does not: its spectrum gives an explicit image of the glassy Gibbs geometry, encoded by the distribution of mutual overlaps between samples. Parisi theory and Monte Carlo confirm this mechanism across simple and glassy phases.

cond-mat.dis-nn↗

arXiv2026

Replica theory for the rate functional of the empirical spectral distribution function of diluted Hermitian matrices

We develop a replica-based framework for the scaled cumulant-generating functional of the empirical spectral distribution function $i_C$ of diluted Hermitian random matrices. Within a replica-symmetric saddle-point assumption, this construction yields a candidate rate functional for fluctuations of $i_C$. As an illustrative application, we consider adjacency matrices of unweighted Erdős-Rényi random graphs with mean degree $c$. We derive explicit expressions for the first two cumulants of $i_C$, indicate how higher cumulants can be obtained from further functional derivatives, and compute the rate function of Fourier coefficients, equivalently of selected linear spectral statistics. The replica-symmetric predictions are tested against exact numerical diagonalization and show good agreement in the accessible fluctuation regime. The approach provides a basis for studying rate functionals of spectral observables in sparse random matrix ensembles.

cond-mat.dis-nn↗

arXiv2026

Dynamical cavity method for continuous-time complex systems on sparse random graphs

Dynamical mean-field theory (DMFT) reduces dense high-dimensional disordered dynamics to a self-consistent effective stochastic process. For sparse and heterogeneous networks, however, local fields contain finitely many strong inputs, so the Gaussian closure mechanisms of dense DMFT need not apply. We develop a continuous-time cavity derivation of sparse-network DMFT at the level of path measures for stochastic dynamics with general pairwise interactions on sparse random graphs. The cavity equations are exact on trees and yield the finite-time thermodynamic description on locally tree-like graphs. They show explicitly how reciprocity changes dynamical closure: fully directed graphs recover the sparse directed path-probability equation, whereas reciprocal or bidirected edges require conditional path kernels driven by the imposed history of the receiving node. Ensemble averaging gives laws over path-probability messages, with barycenters and higher-message moments closing by multilinearity and independence of incoming branches. A causal discrete-time derivation yields the corresponding population-dynamics representation, distinguishing trajectory populations for directed graphs from conditional branch-law or finite-depth tree populations for reciprocal graphs. We also formulate finite-memory numerical closures and test them in an additive-input recurrent neural network specialization. Finally, high-connectivity limits are obtained as projections of the sparse path-measure theory, clarifying when dense drift, noise, and response channels reduce to standard low-dimensional DMFT and when path-level descriptions remain essential.

cond-mat.dis-nn↗

arXiv2026

Statistical Mechanics of Random Matrices

These lecture notes are based on the lectures on \emph{Statistical Mechanics of Random Matrices} delivered at the Spring College on the Physics of Complex Systems, held at the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, from 19 February to 15 March 2024. Their aim is to present a statistical-mechanics route to the spectral theory of sparse and diluted random matrices, with emphasis on cavity and replica methods, resolvent techniques, population dynamics, typical spectral densities, spectral-count fluctuations and large deviations, conditioned spectra, and non-Hermitian extensions. The written form of the notes has been deliberately expanded beyond the material actually covered during the lectures. This is partly because a set of lecture notes can afford a more systematic development than a sequence of blackboard lectures, and partly because several natural continuations of the material become clearer once the central methods have been introduced. Consequently, not every topic discussed here was presented during the College. The additional material is included to give a more coherent account of the subject and to indicate directions that, hopefully, can be covered in greater detail in future lectures or schools. Since these are lecture notes rather than a state-of-the-art review, the choice of topics is necessarily selective and is naturally tilted towards the author's own work and collaborations on this subject. I have nevertheless tried, within the limits of this format, to place the material in contact with the broader literature and to represent the surrounding state of the art as fairly as possible. Inevitably, some relevant contributions may be missing or treated too briefly; such omissions are unintentional and reflect the pedagogical scope of the notes rather than a judgement on their importance.

cond-mat.dis-nn↗

arXiv2026

Benchmarking a restricted Boltzmann machine on the $\mathbb{Z}_2$ Bose-Hubbard chain in the adiabatic hard-core regime

We study the ground state of the $\mathbb{Z}_2$ Bose-Hubbard chain in the adiabatic hard-core limit at half filling using variational Monte Carlo with a shallow restricted Boltzmann machine as the variational ansatz. In this context, the neural quantum state is compared with the established adiabatic description of the model. The variational results reproduce the overall structure of the phase diagram obtained from magnetization observables, distinguish the polarized and Néel-ordered regions, and capture representative spin patterns and site occupations for the staggered insulating configurations selected by a weak symmetry-breaking field. Taken together, these results show that a shallow restricted Boltzmann machine reproduces the main adiabatic phase structure of the one-dimensional $\mathbb{Z}_2$ Bose-Hubbard chain and captures the selected symmetry-broken insulating configurations at half filling.

cond-mat.quant-gas↗

arXiv2026

Experiments, Computability, and the Existence of Physical Functions

Experimental science usually relies on laboratory procedures that, after finitely many steps, terminate with numerical reports on physical quantities. This paper argues that such procedures can be understood as algorithmic once the protocol, background conditions, and reporting rules are fixed. Assuming an explicit physical Church--Turing bridge principle, a reproducible experiment therefore computes a map from admissible inputs to outputs, and the corresponding function exists in the sense appropriate to those outputs. Furthermore, computable analysis allows us to explain why this conclusion is compatible with finite-precision measurement since in this case what matters is a systematic approximation to a requested accuracy, not the production of exact real numbers in a single step. Neither protocol dependence nor stochasticity undermines the existence claim. Rather, they specify which map is realized by a given protocol and what additional assumptions are required for stronger claims about a single protocol-independent quantity. The paper therefore separates three questions that are often conflated: whether the function exists, whether it is computable, and when results obtained under different protocols may be treated as measurements of the same quantity.

physics.hist-ph↗

arXiv2026

Beyond Whittle: exact finite-time multispectral statistics from a single Brownian trajectory in a harmonic trap

Power spectral densities are often interpreted through ensemble averages and long-time asymptotics. In many experiments, however, only a single finite record is available, so spectral estimators remain broadly distributed and the usual independence assumptions across frequencies need not hold. Here we develop an exact finite-$T$ multispectral theory for an overdamped Brownian particle in a harmonic trap. For a collection of frequencies $\{ω_i\}$, we obtain an exact characterization of the joint law of the finite-time estimators $\{S(ω_i,T)\}$, together with a covariance-explicit Gaussian representation for the associated Fourier projections. This representation makes the observation-window-induced inter-frequency correlations explicit and shows how they vanish as $T\to\infty$, thereby recovering the asymptotic Whittle picture. We then use this structure to formulate a hierarchy of spectral likelihoods for inference from a single trajectory, ranging from the factorized Whittle approximation to blockwise covariance-aware approximations in frequency space. Monte Carlo simulations validate the finite-time theory and quantify the effect of neglected cross-frequency correlations on single-trajectory estimates of the trap parameters. Our results provide a controlled finite-time benchmark for spectral inference beyond the asymptotic regime.

cond-mat.stat-mech↗

arXiv2026

Active Learning vs Traditional Lecturing in Introductory Mechanics: A Pooled Pass-Rate Benchmark Under Common Departmental Assessments from a Latin American Institutional Change Initiative

Improving student success in introductory physics remains a persistent challenge despite substantial progress from research-based instructional practices. Evidence from the Latin American context remains limited, where resources for instructional change are often constrained. This study reports a transparent benchmark of student passing outcomes in \textit{Elementary Mechanics I} at a large public university in México, comparing sections using Active Learning (AL) with those using Traditional Lecturing (TL). The labels AL and TL are operational, referring to section-level implementations by individual instructors rather than standardized protocols. Using aggregated counts from coordinator reports and common departmental assessments -- written by a committee independent of instructional modality -- we estimated pooled student-level pass probabilities for the first and second midterm exams, the global exam, and the final mark. Modality differences are summarized primarily by the risk difference, $RD_a=p_{\mathrm{AL},a}-p_{\mathrm{TL},a}$ (percentage points), with uncertainty quantified using Wilson confidence intervals and a Bayesian reference analysis with Jeffreys priors for binomial proportions. Across assessments, pooled pass rates were higher under AL than under TL, with the strongest separation observed for the global exam and the final mark. For these outcomes, the $95\%$ confidence intervals excluded zero, including under a random-intercept Bayesian model. We emphasize a constrained interpretation: the results provide a student-weighted benchmark of ``AL as implemented'' versus ``TL as implemented'' in this setting, without isolating the causal effect of individual instructional techniques. Implications are discussed for departmental decision-making and feasible next steps in evaluation, including improved student data collection and more robust qualitative analysis.

physics.ed-ph↗

arXiv2025

Understanding and improving axial detection in optical tweezers based on the interference of forward- and backward- scattered light

Fast and accurate 3D position detection in optical tweezers (OT) is essential for quantitatively monitoring subtle variations in the mechanical properties of microscopic systems ranging from biomolecules to cells and colloids. Because standard OT configurations do not provide direct access to the axial position, axial detection typically relies on temporal fluctuations in forward-scattered optical power to infer the position of the particle. This approach generally assumes a linear-response regime in which the signal arises from the interference between the forward scattered and the nonscattered optical fields; however, under certain conditions, the backward-scattered contribution becomes non-negligible, leading to deviations from the linear response. Here, we present a simple yet comprehensive model for axial detection in standard OT while explicitly accounting for the backward-scattered field. Together with experimental validation, this framework neatly explains the standing-wave response observed when the backward-scattered field interferes with the nonscattered and the forward-scattered components, enabling accurate estimation of trap stiffness and particle diffusion under more general conditions. This work deepens our understanding of the phenomenology observed in real optical-tweezers measurements and extends their capabilities to conditions where standard approaches fail.

physics.optics↗

arXiv2025

Spectral properties of a Non-Hermitian extension of the diluted Wishart ensemble

We develop a theoretical framework based on the cavity and replica methods to analyze the spectral properties of sparse asymmetric correlation matrices of the form $\boldsymbol{F} = (\boldsymbol{X}\boldsymbol{Y}^\top + ω\boldsymbol{Y}\boldsymbol{X}^\top)/2T$, where $\boldsymbol{X}$ and $\boldsymbol{Y}$ are adjacency matrices of weighted Erdős--Rényi random graphs. We examine how the spectral density evolves as the asymmetry parameter $ω$ varies from $0 < ω< 1$ (nearly symmetric matrices) to $-1 < ω\le 0$ (nearly antisymmetric matrices). Analytical predictions are validated through exact numerical diagonalization, showing excellent agreement with theoretical results in the thermodynamic limit.

cond-mat.dis-nn↗

arXiv2025

Tunable Entangling and Steering of Ferrimagnetic Magnons via an OptoMagnoMechanical Ring

Recently, magnomechanical systems have emerged as promising platforms for quantum technologies, exploiting magnon-photon-phonon interactions to store high-fidelity quantum information. In this paper, we propose a scheme to entangle two spatially separated ferrimagnetic YIG crystals by injecting a microwave field into an optomagnonic ring cavity. The proposed optomagnomechanical configuration utilizes the coupling between magnetostriction-induced mechanical displacements and the optical cavity via radiation pressure. Magnons - collective spin excitations in macroscopic ferromagnets - are directly driven by an electromagnetic field. We demonstrate the generation of macroscopic magnon entanglement by exciting the optical cavity with a red detuned microwave field and the YIG crystals with a blue detuned field. Our analysis reveals that magnon entanglement vanishes for identical magnomechanical couplings but remains robust against thermal fluctuations. The magnon modes entangled in two ferrimagnetic crystals represent genuine macroscopic quantum states with potential applications in the study of macroscopic quantum mechanics and quantum information processing based on magnonics. The configuration is based on experimentally accessible parameters, providing a feasible route of quantum technologies.

quant-ph↗

arXiv2023

Optimal calibration of optical tweezers with arbitrary integration time and sampling frequencies -- A general framework

Optical tweezers (OT) have become an essential technique in several fields of physics, chemistry, and biology as precise micromanipulation tools and microscopic force transducers. Quantitative measurements require the accurate calibration of the trap stiffness of the optical trap and the diffusion constant of the optically trapped particle. This is typically done by statistical estimators constructed from the position signal of the particle, which is recorded by a digital camera or a quadrant photodiode. The finite integration time and sampling frequency of the detector need to be properly taken into account. Here, we present a general approach based on the joint probability density function of the sampled trajectory that corrects exactly the biases due to the detector's finite integration time and limited sampling frequency, providing theoretical formulas for the most widely employed calibration methods: equipartition, mean squared displacement, autocorrelation, power spectral density, and force reconstruction via maximum-likelihood-estimator analysis (FORMA). Our results, tested with experiments and Monte Carlo simulations, will permit users of OT to confidently estimate the trap stiffness and diffusion constant, extending their use to a broader set of experimental conditions.

physics.optics↗

arXiv2022

A smooth transition towards a Tracy-Widom distribution for the largest eigenvalue of interacting $k$-body fermionic Embedded Gaussian Ensembles

In spite of its simplicity, the central limit theorem captures one of the most outstanding phenomena in mathematical physics, that of universality. While this classical result is well understood it is still not very clear what happens to this universal behaviour when the random variables become correlated. A fruitful mathematically laboratory to investigate the rising of new universal properties is offered by the set of eigenvalues of random matrices. In this regard a lot of work has been done using the standard random matrix ensembles and focusing on the distribution of extreme eigenvalues. In this case, the distribution of the largest -- or smallest -- eigenvalue departs from the Fisher-Tippett-Gnedenko theorem yielding the celebrated Tracy-Widom distribution. One may wonder, yet again, how robust is this new universal behaviour captured by the Tracy-Widom distribution when the correlation among eigenvalues changes. Few answers have been provided to this poignant question and our intention in the present work is to contribute to this interesting unexplored territory. Thus, we study numerically the probability distribution for the normalized largest eigenvalue of the interacting $k$-body fermionic orthogonal and unitary Embedded Gaussian Ensembles in the diluted limit. We find a smooth transition from a slightly asymmetric Gaussian-like distribution, for small $k/m$, to the Tracy-Widom distribution as $k/m\to 1$, where $k$ is the rank of the interaction and $m$ is the number of fermions. Correlations at the edge of the spectrum are stronger for small values of $k/m$, and are independent of the number of particles considered. Our results indicate that subtle correlations towards the edge of the spectrum distinguish the statistical properties of the spectrum of interacting many-body systems in the dilute limit, from those expected for the standard random matrix ensembles.

cond-mat.dis-nn↗

arXiv2022

Roadmap for Optical Tweezers

Optical tweezers are tools made of light that enable contactless pushing, trapping, and manipulation of objects ranging from atoms to space light sails. Since the pioneering work by Arthur Ashkin in the 1970s, optical tweezers have evolved into sophisticated instruments and have been employed in a broad range of applications in life sciences, physics, and engineering. These include accurate force and torque measurement at the femtonewton level, microrheology of complex fluids, single micro- and nanoparticle spectroscopy, single-cell analysis, and statistical-physics experiments. This roadmap provides insights into current investigations involving optical forces and optical tweezers from their theoretical foundations to designs and setups. It also offers perspectives for applications to a wide range of research fields, from biophysics to space exploration.

physics.optics↗

arXiv2022

Spectral properties of the generalized diluted Wishart ensemble

The celebrated Marčenko-Pastur law, that considers the asymptotic spectral density of random covariance matrices, has found a great number of applications in physics, biology, economics, engineering, among others. Here, using techniques from statistical mechanics of spin glasses, we derive simple formulas concerning the spectral density of generalized diluted Wishart matrices. These are defined as $\bm{F}\equiv \frac{1}{2d}\left( \bm{X}\bm{Y}^T+ \bm{Y}\bm{X}^T\right)$, where $\bm{X}$ and $\bm{Y}$ are diluted $N\times P$ rectangular matrices, whose entries correspond to the links of doubly-weighted random bipartite Poissonian graphs following the distribution $P(x_{i}^μ,y_{i}^μ)=\frac{d}{N}\varrho(x_{i}^μ,y_{i}^μ)+\left(1-\frac{d}{N}\right)δ_{x_{i}^μ,0}δ_{y_{i}^μ,0}$, with the probability density $\varrho(x,y)$ controlling the correlation between the matrices entries of $\bm{X}$ and $\bm{Y}$. Our results cover several interesting cases by varying the parameters of the matrix ensemble, namely, the dilution of the graph $d$, the rectangularity of the matrices $α=N/P$, and the degree of correlation of the matrix entries via the density $\varrho(x,y)$. Finally, we compare our findings to numerical diagonalisation showing excellent agreement.

cond-mat.dis-nn↗

arXiv2021

Analytic solution of the two-star model with correlated degrees

Exponential random graphs are important to model the structure of real-world complex networks. Here we solve the two-star model with degree-degree correlations in the sparse regime. The model constraints the average correlation between the degrees of adjacent nodes (nearest neighbors) and between the degrees at the end-points of two-stars (next nearest neighbors). We compute exactly the network free energy and show that this model undergoes a first-order transition to a condensed phase. For non-negative degree correlations between next nearest neighbors, the degree distribution inside the condensed phase has a single peak at the largest degree, while for negative degree correlations between next nearest neighbors the condensed phase is characterized by a bimodal degree distribution. We calculate the degree assortativities and show they are non-monotonic functions of the model parameters, with a discontinuous behavior at the first-order transition. The first-order critical line terminates at a second-order critical point, whose location in the phase diagram can be accurately determined. Our results can help to develop more detailed models of complex networks with correlated degrees.

cond-mat.stat-mech↗

arXiv2021

Efficient qubit phase estimation using adaptive measurements

Estimating correctly the quantum phase of a physical system is a central problem in quantum parameter estimation theory due to its wide range of applications from quantum metrology to cryptography. Ideally, the optimal quantum estimator is given by the so-called quantum Cramér-Rao bound, so any measurement strategy aims to obtain estimations as close as possible to it. However, more often than not, the current state-of-the-art methods to estimate quantum phases fail to reach this bound as they rely on maximum likelihood estimators of non-identifiable likelihood functions. In this work we thoroughly review various schemes for estimating the phase of a qubit, identifying the underlying problem which prohibits these methods to reach the quantum Cramér-Rao bound, and propose a new adaptive scheme based on covariant measurements to circumvent this problem. Our findings are carefully checked by Monte Carlo simulations, showing that the method we propose is both mathematically and experimentally more realistic and more efficient than the methods currently available.

quant-ph↗

arXiv2021

Optimal work extraction and mutual information in a generalized Szilárd engine

A 1929 Gedankenexperiment proposed by Szilárd, often referred to as "Szilárd's engine", has served as a foundation for computing fundamental thermodynamic bounds to information processing. While Szilárd's original box could be partitioned into two halves and contains one gas molecule, we calculate here the maximal average work that can be extracted in a system with $N$ particles and $q$ partitions, given an observer which counts the molecules in each partition, and given a work extraction mechanism that is limited to pressure equalization. We find that the average extracted work is proportional to the mutual information between the one-particle position and the vector containing the counts of how many particles are in each partition. We optimize this quantity over the initial locations of the dividing walls, and find that there exists a critical number of particles $N^{\star}(q)$ below which the extracted work is maximized by a symmetric configuration of the $q$ partitions, and above which the optimal partitioning is asymmetric. Overall, the average extracted work is maximized for a number of particles $\hat{N}(q)<N^{\star}(q)$, with a symmetric partition. We calculate asymptotic values for $N\rightarrow \infty$.

cond-mat.stat-mech↗