scholarly journals Airy Phenomena and Analytic Combinatorics of Connected Graphs

10.37236/1787 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Philippe Flajolet ◽  
Bruno Salvy ◽  
Gilles Schaeffer
Keyword(s):  
Saddle Point ◽  
Saddle Points ◽  
Formal Series ◽  
Probabilistic Methods ◽  
Saddle Point Method ◽  
Airy Functions ◽  
Connected Graphs ◽  
Analytic Combinatorics ◽  
Point Analysis ◽  
Combinatorial Decompositions

Until now, the enumeration of connected graphs has been dealt with by probabilistic methods, by special combinatorial decompositions or by somewhat indirect formal series manipulations. We show here that it is possible to make analytic sense of the divergent series that expresses the generating function of connected graphs. As a consequence, it becomes possible to derive analytically known enumeration results using only first principles of combinatorial analysis and straight asymptotic analysis—specifically, the saddle-point method. In this perspective, the enumeration of connected graphs by excess (of number of edges over number of vertices) derives from a simple saddle-point analysis. Furthermore, a refined analysis based on coalescent saddle points yields complete asymptotic expansions for the number of graphs of fixed excess, through an explicit connection with Airy functions.

scholarly journals Phase profile of the wave function of canonical tensor model and emergence of large space–times

2021 ◽  
Author(s):  
Naoki Sasakura
Keyword(s):  
Wave Function ◽  
Saddle Point ◽  
Saddle Points ◽  
Space Time ◽  
Tensor Rank ◽  
Tensor Model ◽  
Large Space ◽  
Time Dynamics ◽  
Point Analysis ◽  
Phase Profile

In this paper, to understand space–time dynamics in the canonical tensor model of quantum gravity for the positive cosmological constant case, we analytically and numerically study the phase profile of its exact wave function in a coordinate representation, instead of the momentum representation analyzed so far. A saddle point analysis shows that Lie group symmetric space–times are strongly favored due to abundance of continuously existing saddle points, giving an emergent fluid picture. The phase profile suggests that spatial sizes grow in “time,” where sizes are measured by the tensor-geometry correspondence previously introduced using tensor rank decomposition. Monte Carlo simulations are also performed for a few small N cases by applying a re-weighting procedure to an oscillatory integral which expresses the wave function. The results agree well with the saddle point analysis, but the phase profile is subject to disturbances in a large space–time region, suggesting existence of light modes there and motivating future computations of primordial fluctuations from the perspective of canonical tensor model.

The saddle-point method in ${\mathbb C}^N$ and the generalized Airy functions

2019 ◽  
Vol 147 (2) ◽  
pp. 221-257
Author(s):  
Francesco PINNA ◽  
Carlo VIOLA
Keyword(s):  
Saddle Point ◽  
Point Method ◽  
Saddle Point Method ◽  
Airy Functions

scholarly journals Joint String Complexity for Markov Sources

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Philippe Jacquet ◽  
Wojciech Szpankowski
Keyword(s):  
Saddle Point ◽  
Saddle Points ◽  
Singularity Analysis ◽  
Point Method ◽  
Saddle Point Method ◽  
Genome Sequences ◽  
Markov Source ◽  
International Audience ◽  
The Common ◽  
Markov Sources

International audience String complexity is defined as the cardinality of a set of all distinct words (factors) of a given string. For two strings, we define $\textit{joint string complexity}$ as the set of words that are common to both strings. We also relax this definition and introduce $\textit{joint semi-complexity}$ restricted to the common words appearing at least twice in both strings. String complexity finds a number of applications from capturing the richness of a language to finding similarities between two genome sequences. In this paper we analyze joint complexity and joint semi-complexity when both strings are generated by a Markov source. The problem turns out to be quite challenging requiring subtle singularity analysis and saddle point method over infinity many saddle points leading to novel oscillatory phenomena with single and double periodicities.

scholarly journals Asymptotics of the Stirling numbers of the second kind revisited

2013 ◽  
Vol 7 (2) ◽  
pp. 193-210 ◽  
Author(s):  
Guy Louchard
Keyword(s):  
Saddle Point ◽  
Generating Function ◽  
Central Region ◽  
Integral Formula ◽  
Stirling Numbers ◽  
Saddle Point Method ◽  
Asymptotic Results ◽  
Analytic Combinatorics ◽  
Central Regions ◽  
Cauchy’S Integral Formula

Using the Saddle point method and multiseries expansions, we obtain from the generating function of the Stirling numbers of the second kind {n / m} and Cauchy's integral formula, asymptotic results in central and non-central regions. In the central region, we revisit the celebrated Gaussian theorem with more precision. In the region m = n - na, 1 > a > 1/2, we analyze the dependence of {n / m} on a. An extension of some Moser and Wyman's result to full m range is also provided. This paper fits within the framework of Analytic Combinatorics.

scholarly journals Existence Theorems ofε-Cone Saddle Points for Vector-Valued Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Tao Chen
Keyword(s):  
Saddle Point ◽  
Existence Theorem ◽  
Equilibrium Problem ◽  
Existence Result ◽  
Existence Theorems ◽  
Saddle Points ◽  
Vector Equilibrium Problem ◽  
Saddle Point Theorem ◽  
Vector Equilibrium ◽  
Vector Valued

A new existence result ofε-vector equilibrium problem is first obtained. Then, by using the existence theorem ofε-vector equilibrium problem, a weaklyε-cone saddle point theorem is also obtained for vector-valued mappings.

Hierarchical Clustering of Dynamical Networks Using a Saddle-Point Analysis

2013 ◽  
Vol 58 (1) ◽  
pp. 113-124 ◽  
Author(s):  
Mathias Burger ◽  
Daniel Zelazo ◽  
Frank Allgower
Keyword(s):  
Saddle Point ◽  
Hierarchical Clustering ◽  
Dynamical Networks ◽  
Point Analysis

scholarly journals A Note on the Asymptotic Behavior of the Heights in $b$-Tries for $b$ Large

10.37236/1517 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Charles Knessl ◽  
Wojciech Szpankowski
Keyword(s):  
Asymptotic Behavior ◽  
Saddle Point ◽  
Generating Functions ◽  
Single Point ◽  
Extreme Value ◽  
Limiting Distribution ◽  
Point Method ◽  
Saddle Point Method ◽  
Numerical Verification ◽  
Height Distribution

We study the limiting distribution of the height in a generalized trie in which external nodes are capable to store up to $b$ items (the so called $b$-tries). We assume that such a tree is built from $n$ random strings (items) generated by an unbiased memoryless source. In this paper, we discuss the case when $b$ and $n$ are both large. We shall identify five regions of the height distribution that should be compared to three regions obtained for fixed $b$. We prove that for most $n$, the limiting distribution is concentrated at the single point $k_1=\lfloor \log_2 (n/b)\rfloor +1$ as $n,b\to \infty$. We observe that this is quite different than the height distribution for fixed $b$, in which case the limiting distribution is of an extreme value type concentrated around $(1+1/b)\log_2 n$. We derive our results by analytic methods, namely generating functions and the saddle point method. We also present some numerical verification of our results.

scholarly journals The Saddle Point Method for the Integral of the Absolute Value of the Brownian Motion

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Leonid Tolmatz
Keyword(s):  
Distribution Function ◽  
Brownian Motion ◽  
Saddle Point ◽  
Point Method ◽  
Saddle Point Method ◽  
Tail Asymptotics ◽  
Absolute Value ◽  
Exact Tail Asymptotics ◽  
International Audience ◽  
The Absolute

International audience The distribution function of the integral of the absolute value of the Brownian motion was expressed by L.Takács in the form of various series. In the present paper we determine the exact tail asymptotics of this distribution function. The proposed method is applicable to a variety of other Wiener functionals as well.

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