scholarly journals Boolean cumulants and subordination in free probability

2020 ◽  
pp. 2150036
Author(s):  
Franz Lehner ◽  
Kamil Szpojankowski
Keyword(s):  
Free Probability ◽  
Borel Function ◽  
General Setting ◽  
Main Tool ◽  
Explicit Calculation ◽  
Conditional Expectations ◽  
Multiplicative Convolution ◽  
Noncommutative Polynomials ◽  
Free Random Variables ◽  
Free Convolutions

Subordination is the basis of the analytic approach to free additive and multiplicative convolution. We extend this approach to a more general setting and prove that the conditional expectation [Formula: see text] for free random variables [Formula: see text] and a Borel function [Formula: see text] is a resolvent again. This result allows the explicit calculation of the distribution of noncommutative polynomials of the form [Formula: see text]. The main tool is a new combinatorial formula for conditional expectations in terms of Boolean cumulants and a corresponding analytic formula for conditional expectations of resolvents, generalizing subordination formulas for both additive and multiplicative free convolutions. In the final section, we illustrate the results with step by step explicit computations and an exposition of all necessary ingredients.

scholarly journals Conditional Expectation, Entropy, and Transport for Convex Gibbs Laws in Free Probability

2020 ◽  
Author(s):  
David Jekel
Keyword(s):  
Matrix Models ◽  
Random Matrix ◽  
Free Probability ◽  
Free Entropy ◽  
Large N ◽  
Conditional Expectations ◽  
The Matrix ◽  
Random Matrix Models ◽  
Regular Convex ◽  
The Law

Abstract Let $(X_1,\dots ,X_m)$ be self-adjoint noncommutative random variables distributed according to the free Gibbs law given by a sufficiently regular convex and semi-concave potential $V$, and let $(S_1,\dots ,S_m)$ be a free semicircular family. For $k < m$, we show that conditional expectations and conditional non-microstates free entropy given $X_1$, …, $X_k$ arise as the large $N$ limit of the corresponding conditional expectations and entropy for the $N \times N$ random matrix models associated to $V$. Then, by studying conditional transport of measure for the matrix models, we construct an isomorphism $\mathrm{W}^*(X_1,\dots ,X_m) \to \mathrm{W}^*(S_1,\dots ,S_m)$ that maps $\mathrm{W}^*(X_1,\dots ,X_k)$ to $\mathrm{W}^*(S_1,\dots ,S_k)$ for each $k = 1, \dots , m$ and that also witnesses the Talagrand inequality for the law of $(X_1,\dots ,X_m)$ relative to the law of $(S_1,\dots ,S_m)$.

scholarly journals Simply Generated Non-Crossing Partitions

2017 ◽  
Vol 26 (4) ◽  
pp. 560-592 ◽  
Author(s):  
IGOR KORTCHEMSKI ◽  
CYRIL MARZOUK
Keyword(s):  
Probability Measure ◽  
Real Line ◽  
Simple Formula ◽  
Limit Theorems ◽  
Block Structure ◽  
Free Probability ◽  
Main Tool ◽  
Plane Trees ◽  
Simply Generated Trees ◽  
Block Sizes

We introduce and study the model of simply generated non-crossing partitions, which are, roughly speaking, chosen at random according to a sequence of weights. This framework encompasses the particular case of uniform non-crossing partitions with constraints on their block sizes. Our main tool is a bijection between non-crossing partitions and plane trees, which maps such simply generated non-crossing partitions into simply generated trees so that blocks of sizekare in correspondence with vertices of out-degreek. This allows us to obtain limit theorems concerning the block structure of simply generated non-crossing partitions. We apply our results in free probability by giving a simple formula relating the maximum of the support of a compactly supported probability measure on the real line in terms of its free cumulants.

scholarly journals Hopf algebras and the logarithm of the S-transform in free probability ― Extended abstract

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Mitja Mastnak ◽  
Alexandru Nica
Keyword(s):  
Hopf Algebras ◽  
Free Probability ◽  
Algebraic Approach ◽  
S Transform ◽  
Multiplicative Convolution ◽  
Free Multiplicative Convolution ◽  
International Audience

International audience This document is an extended abstract of the paper `Hopf algebras and the logarithm of the S-transform in free probability' in which we introduce a Hopf algebraic approach to the study of the operation $\boxtimes$ (free multiplicative convolution) from free probability.

scholarly journals Multivariate Fuss-Narayana Polynomials and their Application to Random Matrices

10.37236/2799 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Romuald Lenczewski ◽  
Rafal Salapata
Keyword(s):  
Random Matrices ◽  
Shape Parameter ◽  
Free Probability ◽  
Homogeneous Polynomials ◽  
Noncrossing Partitions ◽  
Multiplicative Convolution ◽  
Free Multiplicative Convolution

It has been shown recently that the limit moments of $W(n)=B(n)B^{*}(n)$, where $B(n)$ is a product of $p$ independent rectangular random matrices, are certain homogeneous polynomials $P_{k}(d_0,d_1, \ldots , d_{p})$ in the asymptotic dimensions of these matrices. Using the combinatorics of noncrossing partitions, we explicitly determine these polynomials and show that they are closely related to polynomials which can be viewed as {\it multivariate Fuss-Narayana polynomials}. Using this result, we compute the moments of $\varrho_{t_1}\boxtimes \varrho_{t_2}\boxtimes\ldots \boxtimes \varrho_{t_m}$ for any positive $t_1,t_2, \ldots , t_m$, where $\boxtimes$ is the free multiplicative convolution in free probability and $\varrho_{t}$ is the Marchenko-Pastur distribution with shape parameter $t$.

Multi-parameter Singular Integrals. (AM-189)

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Partial Differential Equations ◽  
Graduate Students ◽  
General Theory ◽  
Differential Equations ◽  
Singular Integrals ◽  
Main Tool ◽  
Classical Theorem ◽  
Quantitative Version ◽  
Linear Partial Differential Equations

This book develops a new theory of multi-parameter singular integrals associated with Carnot–Carathéodory balls. The book first details the classical theory of Calderón–Zygmund singular integrals and applications to linear partial differential equations. It then outlines the theory of multi-parameter Carnot–Carathéodory geometry, where the main tool is a quantitative version of the classical theorem of Frobenius. The book then gives several examples of multi-parameter singular integrals arising naturally in various problems. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples. This is one of the first general theories of multi-parameter singular integrals that goes beyond the product theory of singular integrals and their analogs. This book will interest graduate students and researchers working in singular integrals and related fields.

Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

2013 ◽  
Author(s):  
Friedhelm Waldhausen ◽  
Bjørn Jahren ◽  
John Rognes
Keyword(s):  
Main Tool ◽  
Homotopy Equivalence ◽  
Classifying Space ◽  
Deformation Retract ◽  
Simplicial Sets ◽  
Full Proof

Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.

The Novel TEM Sample Preparation Approach for Targeted via with Barrier/Cu Seed Layer Inspection

2009 ◽  
Author(s):  
Wen-Fei Hsieh ◽  
Shih-Hsiang Tseng ◽  
Bo Min She
Keyword(s):  
Sample Preparation ◽  
Cross Section ◽  
Seed Layer ◽  
Target Location ◽  
Process Development ◽  
Process Integration ◽  
Ion Beam ◽  
Main Tool ◽  
Dual Beam ◽  
Sample Preparation Procedure

Abstract In this study, an FIB-based cross section TEM sample preparation procedure for targeted via with barrier/Cu seed layer is introduced. The dual beam FIB with electron beam for target location and Ga ion beam for sample milling is the main tool for the targeted via with barrier/Cu seed layer inspection. With the help of the FIB operation and epoxy layer protection, ta cross section TEM sample at a targeted via with barrier/Cu seed layer could be made. Subsequent TEM inspection is used to verify the quality of the structure. This approach was used in the Cu process integration performance monitor. All these TEM results are very helpful in process development and yield improvement.

To the discussion about the modern world order

2020 ◽  
pp. 35-41
Author(s):  
A. Mustafabeyli
Keyword(s):  
Soviet Union ◽  
Intergovernmental Relations ◽  
World System ◽  
Main Tool ◽  
World Order ◽  
Modern World ◽  
The Soviet Union ◽  
The World ◽  
And Behavior ◽  
Second World

In many political researches there if a conclusion that the world system which was founded after the Second world war is destroyed of chaos. But the world system couldn`t work while the two opposite systems — socialist and capitalist were in hard confrontation. After collapse of the Soviet Union and the European socialist community the nature of intergovernmental relations and behavior of the international community did not change. The power always was and still is the main tool of international communication.

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