The Characteristic Polynomial of a Random Permutation Matrix at Different Points

We consider the logarithm of the characteristic polynomial of random permutation matrices, evaluated on a finite set of different points. The permutations are chosen with respect to the Ewens distribution on the symmetric group. We show that the behavior at different points is independent in the limit and are asymptotically normal. Our methods enables us to study more general matrices, closely related to permutation matrices, and multiplicative class functions.Comment: 30 pages, 2 figures. Differences to Version 1: We have improved the presentation and add some references Stochastic Processes and their Applications, 201

Central limit theorem for multiplicative class functions on the symmetric group

Hambly, Keevash, O'Connell and Stark have proven a central limit theorem for the characteristic polynomial of a permutation matrix with respect to the uniform measure on the symmetric group. We generalize this result in several ways. We prove here a central limit theorem for multiplicative class functions on symmetric group with respect to the Ewens measure and compute the covariance of the real and the imaginary part in the limit. We also estimate the rate of convergence with the Wasserstein distance.Comment: 23 pages; the mathematics is the same as in the previous version, but there are several improvments in the presentation, including a more intuitve name for the considered function

Random permutations with logarithmic cycle weights

We consider random permutations on Sn with logarithmic growing cycles weights and study the asymptotic behavior as the length n tends to infinity. We show that the cycle count process converges to a vector of independent Poisson variables and also compute the total variation distance between both processes. Next, we prove a central limit theorem for the total number of cycles. Furthermore we establish a shape theorem and a functional central limit theorem for the Young diagrams associated to random permutations under this measure. We prove these results using tools from complex analysis and combinatorics. In particular we have to apply the method of singularity analysis to generating functions of the form exp((− log(1 − z))k+1) with k ≥ 1, which have not yet been studied in the literature

Does low and oscillatory wall shear stress correlate spatially with early atherosclerosis? A systematic review

Low and oscillatory wall shear stress is widely assumed to play a key role in the initiation and development of atherosclerosis. Indeed, some studies have relied on the low shear theory when developing diagnostic and treatment strategies for cardiovascular disease. We wished to ascertain if this consensus is justified by published data. We performed a systematic review of papers that compare the localization of atherosclerotic lesions with the distribution of haemodynamic indicators calculated using computational fluid dynamics. The review showed that although many articles claim their results conform to the theory, it has been interpreted in different ways: a range of metrics has been used to characterize the distribution of disease, and they have been compared with a range of haemodynamic factors. Several studies, including all of those making systematic point-by-point comparisons of shear and disease, failed to find the expected relation. The various pre- and post-processing techniques used by different groups have reduced the range of shears over which correlations were sought, and in some cases are mutually incompatible. Finally, only a subset of the known patterns of disease has been investigated. The evidence for the low/oscillatory shear theory is less robust than commonly assumed. Longitudinal studies starting from the healthy state, or the collection of average flow metrics derived from large numbers of healthy vessels, both in conjunction with point-by-point comparisons using appropriate statistical techniques, will be necessary to improve our understanding of the relation between blood flow and atherogenesis

Oncoplastic Breast Consortium consensus conference on nipple-sparing mastectomy

Purpose Indications for nipple-sparing mastectomy (NSM) have broadened to include the risk reducing setting and locally advanced tumors, which resulted in a dramatic increase in the use of NSM. The Oncoplastic Breast Consortium consensus conference on NSM and immediate reconstruction was held to address a variety of questions in clinical practice and research based on published evidence and expert panel opinion. Methods The panel consisted of 44 breast surgeons from 14 countries across four continents with a background in gynecology, general or reconstructive surgery and a practice dedicated to breast cancer, as well as a patient advocate. Panelists presented evidence summaries relating to each topic for debate during the in-person consensus conference. The iterative process in question development, voting, and wording of the recommendations followed the modified Delphi methodology. Results Consensus recommendations were reached in 35, majority recommendations in 24, and no recommendations in the remaining 12 questions. The panel acknowledged the need for standardization of various aspects of NSM and immediate reconstruction. It endorsed several oncological contraindications to the preservation of the skin and nipple. Furthermore, it recommended inclusion of patients in prospective registries and routine assessment of patient-reported outcomes. Considerable heterogeneity in breast reconstruction practice became obvious during the conference. Conclusions In case of conflicting or missing evidence to guide treatment, the consensus conference revealed substantial disagreement in expert panel opinion, which, among others, supports the need for a randomized trial to evaluate the safest and most efficacious reconstruction techniques

Oncoplastic Breast Consortium consensus conference on nipple-sparing mastectomy.

Purpose Indications for nipple-sparing mastectomy (NSM) have broadened to include the risk reducing setting and locally advanced tumors, which resulted in a dramatic increase in the use of NSM. The Oncoplastic Breast Consortium consensus conference on NSM and immediate reconstruction was held to address a variety of questions in clinical practice and research based on published evidence and expert panel opinion. Methods The panel consisted of 44 breast surgeons from 14 countries across four continents with a background in gynecology, general or reconstructive surgery and a practice dedicated to breast cancer, as well as a patient advocate. Panelists presented evidence summaries relating to each topic for debate during the in-person consensus conference. The iterative process in question development, voting, and wording of the recommendations followed the modified Delphi methodology. Results Consensus recommendations were reached in 35, majority recommendations in 24, and no recommendations in the remaining 12 questions. The panel acknowledged the need for standardization of various aspects of NSM and immediate reconstruction. It endorsed several oncological contraindications to the preservation of the skin and nipple. Furthermore, it recommended inclusion of patients in prospective registries and routine assessment of patient-reported outcomes. Considerable heterogeneity in breast reconstruction practice became obvious during the conference. Conclusions In case of conflicting or missing evidence to guide treatment, the consensus conference revealed substantial disagreement in expert panel opinion, which, among others, supports the need for a randomized trial to evaluate the safest and most efficacious reconstruction techniques

On the number of eigenvalues of modified permutation matrices in mesoscopic intervals

International audienceWe are interested in two random matrix ensembles related to permutations: the ensemble of permutation matrices following Ewens' distribution of a given parameter $\theta >0$, and its modification where entries equal to $1$ in the matrices are replaced by independent random variables uniformly distributed on the unit circle. For the elements of each ensemble, we focus on the random numbers of eigenvalues lying in some specified arcs of the unit circle. We show that for a finite number of fixed arcs, the fluctuation of the numbers of eigenvalues belonging to them is asymptotically Gaussian. Moreover, for a single arc, we extend this result to the case where the length goes to zero sufficiently slowly when the size of the matrix goes to infinity. Finally, we investigate the behaviour of the largest and smallest spacing between two distinct consecutive eigenvalues