scholarly journals A Coupling of the Spectral Measures at a Vertex

10.37236/8674 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Thibault Espinasse ◽  
Paul Rochet
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Point Of View ◽  
Spectral Measures ◽  
Star Graphs ◽  
Combinatorial Properties ◽  
Heaps Of Pieces ◽  
Generalized Moments

Given the adjacency matrix of an undirected graph, we define a coupling of the spectral measures at the vertices, whose moments count the rooted closed paths in the graph. The resulting joint spectral measure verifies numerous interesting properties that allow to recover minors of analytic functions of the adjacency matrix from its generalized moments. We prove an extension of Obata’s Central Limit Theorem in growing star-graphs to the multivariate case and discuss some combinatorial properties using Viennot’s heaps of pieces point of view.

scholarly journals MONOTONE INDEPENDENCE, COMB GRAPHS AND BOSE–EINSTEIN CONDENSATION

2004 ◽  
Vol 07 (03) ◽  
pp. 419-435 ◽  
Author(s):  
LUIGI ACCARDI ◽  
ANIS BEN GHORBAL ◽  
NOBUAKI OBATA
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Adjacency Matrix ◽  
Spectral Distribution ◽  
Vacuum State ◽  
Independent Random Variables ◽  
Explicit Calculation ◽  
Einstein Condensation ◽  
Bose Einstein Condensation ◽  
Bose Einstein

The adjacency matrix of a comb graph is decomposed into a sum of monotone independent random variables with respect to the vacuum state. The vacuum spectral distribution is shown to be asymptotically the arcsine law as a consequence of the monotone central limit theorem. As an example the comb lattice is studied with explicit calculation.

CONVOLUTION AND CENTRAL LIMIT THEOREM ARISING FROM ADDITION OF FIELD OPERATORS IN ONE-MODE TYPE INTERACTING FOCK SPACES

2005 ◽  
Vol 08 (04) ◽  
pp. 651-657 ◽  
Author(s):  
ANNA DOROTA KRYSTEK ◽  
ŁUKASZ JAN WOJAKOWSKI
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Fock Spaces ◽  
Spectral Measures ◽  
Recurrence Coefficients

In Ref. 2 the authors introduced field operators in one-mode type Interacting Fock Spaces whose spectral measures have common symmetric Jacobi recurrence coefficients but differ in the nonsymmetric ones. We show that the convolution of measures arising from addition of such field operators is the universal convolution of Accardi and Bożejko. We also present the associated central limit theorem in a more general form than in Ref. 2 and give it a proof based on the properties of the convolution.

Depths in hooking networks

2021 ◽  
pp. 1-9
Author(s):  
Colin Desmarais ◽  
Hosam Mahmoud
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Preferential Attachment ◽  
Building Blocks ◽  
Point Of View ◽  
Random Choice ◽  
Wide Range ◽  
Affinity Model ◽  
The One ◽  
Attachment Mechanism

Abstract A hooking network is built by stringing together components randomly chosen from a set of building blocks (graphs with hooks). The vertices are endowed with “affinities” which dictate the attachment mechanism. We study the distance from the master hook to a node in the network chosen according to its affinity after many steps of growth. Such a distance is commonly called the depth of the chosen node. We present an exact average result and a rather general central limit theorem for the depth. The affinity model covers a wide range of attachment mechanisms, such as uniform attachment and preferential attachment, among others. Naturally, the limiting normal distribution is parametrized by the structure of the building blocks and their probabilities. We also take the point of view of a visitor uninformed about the affinity mechanism by which the network is built. To explore the network, such a visitor chooses the nodes uniformly at random. We show that the distance distribution under such a uniform choice is similar to the one under random choice according to affinities.

scholarly journals On the Markov Transition Kernels for First Passage Percolation on the Ladder

2011 ◽  
Vol 48 (02) ◽  
pp. 366-388 ◽  
Author(s):  
Eckhard Schlemm
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Almost Sure Convergence ◽  
First Passage Percolation ◽  
First Passage ◽  
Percolation Problem ◽  
Vertex Set ◽  
Step Transition

We consider the first passage percolation problem on the random graph with vertex set N x {0, 1}, edges joining vertices at a Euclidean distance equal to unity, and independent exponential edge weights. We provide a central limit theorem for the first passage times l n between the vertices (0, 0) and (n, 0), thus extending earlier results about the almost-sure convergence of l n / n as n → ∞. We use generating function techniques to compute the n-step transition kernels of a closely related Markov chain which can be used to explicitly calculate the asymptotic variance in the central limit theorem.

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