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 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.

scholarly journals Sommes friables d'exponentielles et applications

2015 ◽  
Vol 67 (3) ◽  
pp. 597-638 ◽  
Author(s):  
Sary Drappeau
Keyword(s):  
Asymptotic Formula ◽  
Saddle Point ◽  
Exponential Sums ◽  
Prime Factor ◽  
Character Sums ◽  
Contour Integration ◽  
Point Method ◽  
Saddle Point Method ◽  
Greatest Prime Factor

AbstractAn integer is said to be y–friable if its greatest prime factor is less than y. In this paper, we obtain estimates for exponential sums over y–friable numbers up to x which are non–trivial when y ≥ . As a consequence, we obtain an asymptotic formula for the number of y-friable solutions to the equation a + b = c which is valid unconditionally under the same assumption. We use a contour integration argument based on the saddle point method, as developped in the context of friable numbers by Hildebrand and Tenenbaum, and used by Lagarias, Soundararajan and Harper to study exponential and character sums over friable numbers.

scholarly journals Power partitions and saddle-point method

2019 ◽  
Vol 204 ◽  
pp. 435-445 ◽  
Author(s):  
Gérald Tenenbaum ◽  
Jie Wu ◽  
Ya-Li Li
Keyword(s):  
Saddle Point ◽  
Point Method ◽  
Saddle Point Method

MATRIX MODELS AND MARGINAL OPERATORS IN THE PLANAR LIMIT

1990 ◽  
Vol 05 (24) ◽  
pp. 1927-1932 ◽  
Author(s):  
GIOVANNI M. CICUTA ◽  
EMILIO MONTALDI
Keyword(s):  
Saddle Point ◽  
Matrix Models ◽  
Interaction Potential ◽  
Point Method ◽  
Saddle Point Method ◽  
Planar Limit

The role of monomials which are power of traces of matrices in matrix models with polynomial interaction potential is examined. Because of well-known factorization properties, the density of eigenvalues is trivially obtained in the planar limit by the saddle point method. Different susceptibilities have positive or negative critical coefficients.

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