Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution

We exploit a relation between the mean number $N_{m}$ of minima of random Gaussian surfaces and extreme eigenvalues of random matrices to understand the critical behaviour of $N_{m}$ in the simplest glass-like transition occuring in a toy model of a single particle in $N$-dimensional random environment, with $N\gg 1$. Varying the control parameter $\mu$ through the critical value $\mu_c$ we analyse in detail how $N_{m}(\mu)$ drops from being exponentially large in the glassy phase to $N_{m}(\mu)\sim 1$ on the other side of the transition. We also extract a subleading behaviour of $N_{m}(\mu)$ in both glassy and simple phases. The width $\delta{\mu}/\mu_c$ of the critical region is found to scale as $N^{-1/3}$ and inside that region $N_{m}(\mu)$ converges to a limiting shape expressed in terms of the Tracy-Widom distribution

Statistical distribution of quantum entanglement for a random bipartite state

We compute analytically the statistics of the Renyi and von Neumann entropies (standard measures of entanglement), for a random pure state in a large bipartite quantum system. The full probability distribution is computed by first mapping the problem to a random matrix model and then using a Coulomb gas method. We identify three different regimes in the entropy distribution, which correspond to two phase transitions in the associated Coulomb gas. The two critical points correspond to sudden changes in the shape of the Coulomb charge density: the appearance of an integrable singularity at the origin for the first critical point, and the detachement of the rightmost charge (largest eigenvalue) from the sea of the other charges at the second critical point. Analytical results are verified by Monte Carlo numerical simulations. A short account of some of these results appeared recently in Phys. Rev. Lett. {\bf 104}, 110501 (2010).Comment: 7 figure

Nonintersecting Brownian interfaces and Wishart random matrices.

We study a system of N nonintersecting (1+1)-dimensional fluctuating elastic interfaces ("vicious bridges") at thermal equilibrium, each subject to periodic boundary condition in the longitudinal direction and in presence of a substrate that induces an external confining potential for each interface. We show that, for a large system and with an appropriate choice of the external confining potential, the joint distribution of the heights of the N nonintersecting interfaces at a fixed point on the substrate can be mapped to the joint distribution of the eigenvalues of a Wishart matrix of size N with complex entries (Dyson index beta=2), thus providing a physical realization of the Wishart matrix. Exploiting this analogy to random matrix, we calculate analytically (i) the average density of states of the interfaces, (ii) the height distribution of the uppermost and lowermost interfaces (extrema), and (iii) the asymptotic (large-N) distribution of the center of mass of the interfaces. In the last case, we show that the probability density of the center of mass has an essential singularity around its peak, which is shown to be a direct consequence of a phase transition in an associated Coulomb gas problem

A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix

2 figuresInternational audienceIn this paper, we first briefly review some recent results on the distribution of the maximal eigenvalue of a $(N\times N)$ random matrix drawn from Gaussian ensembles. Next we focus on the Gaussian Unitary Ensemble (GUE) and by suitably adapting a method of orthogonal polynomials developed by Gross and Matytsin in the context of Yang-Mills theory in two dimensions, we provide a rather simple derivation of the Tracy-Widom law for GUE. Our derivation is based on the elementary asymptotic scaling analysis of a pair of coupled nonlinear recursion relations. As an added bonus, this method also allows us to compute the precise subleading terms describing the right large deviation tail of the maximal eigenvalue distribution. In the Yang-Mills language, these subleading terms correspond to non-perturbative (in $1/N$ expansion) corrections to the two-dimensional partition function in the so called 'weak' coupling regime

A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix

2 figuresInternational audienceIn this paper, we first briefly review some recent results on the distribution of the maximal eigenvalue of a $(N\times N)$ random matrix drawn from Gaussian ensembles. Next we focus on the Gaussian Unitary Ensemble (GUE) and by suitably adapting a method of orthogonal polynomials developed by Gross and Matytsin in the context of Yang-Mills theory in two dimensions, we provide a rather simple derivation of the Tracy-Widom law for GUE. Our derivation is based on the elementary asymptotic scaling analysis of a pair of coupled nonlinear recursion relations. As an added bonus, this method also allows us to compute the precise subleading terms describing the right large deviation tail of the maximal eigenvalue distribution. In the Yang-Mills language, these subleading terms correspond to non-perturbative (in $1/N$ expansion) corrections to the two-dimensional partition function in the so called 'weak' coupling regime

Phase Transitions in the Distribution of Bipartite Entanglement of a Random Pure State

4 pages revtex, 2 figures includedInternational audienceUsing a Coulomb gas method, we compute analytically the probability distribution of the Renyi entropies (a standard measure of entanglement) for a random pure state of a large bipartite quantum system. We show that, for any order q>1 of the Renyi entropy, there are two critical values at which the entropy's probability distribution changes shape. These critical points correspond to two different transitions in the corresponding charge density of the Coulomb gas: the disappearance of an integrable singularity at the origin and the detachement of a single-charge drop from the continuum sea of all the other charges. These transitions respectively control the left and right tails of the entropy's probability distribution, as verified also by Monte Carlo numerical simulations of the Coulomb gas equilibrium dynamics

A definitive prognostication system for patients with thoracic malignancies diagnosed with COVID-19: an update from the TERAVOLT registry

Background: Patients with thoracic malignancies are at increased risk for mortality from Coronavirus disease 2019 (COVID-19) and large number of intertwined prognostic variables have been identified so far. Methods: Capitalizing data from the TERAVOLT registry, a global study created with the aim of describing the impact of COVID-19 in patients with thoracic malignancies, we used a clustering approach, a fast-backward step-down selection procedure and a tree-based model to screen and optimize a broad panel of demographics, clinical COVID-19 and cancer characteristics. Results: As of April 15, 2021, 1491 consecutive evaluable patients from 18 countries were included in the analysis. With a mean observation period of 42 days, 361 events were reported with an all-cause case fatality rate of 24.2%. The clustering procedure screened approximately 73 covariates in 13 clusters. A further multivariable logistic regression for the association between clusters and death was performed, resulting in five clusters significantly associated with the outcome. The fast-backward step-down selection then identified seven major determinants of death ECOG-PS (OR 2.47 1.87-3.26), neutrophil count (OR 2.46 1.76-3.44), serum procalcitonin (OR 2.37 1.64-3.43), development of pneumonia (OR 1.95 1.48-2.58), c-reactive protein (CRP) (OR 1.90 1.43-2.51), tumor stage at COVID-19 diagnosis (OR 1.97 1.46-2.66) and age (OR 1.71 1.29-2.26). The ROC analysis for death of the selected model confirmed its diagnostic ability (AUC 0.78; 95%CI: 0.75 - 0.81). The nomogram was able to classify the COVID-19 mortality in an interval ranging from 8% to 90% and the tree-based model recognized ECOG-PS, neutrophil count and CRP as the major determinants of prognosis. Conclusion: From 73 variables analyzed, seven major determinants of death have been identified. Poor ECOG-PS demonstrated the strongest association with poor outcome from COVID-19. With our analysis we provide clinicians with a definitive prognostication system to help determine the risk of mortality for patients with thoracic malignancies and COVID-19

Effect of anakinra versus usual care in adults in hospital with COVID-19 and mild-to-moderate pneumonia (CORIMUNO-ANA-1): a randomised controlled trial

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