## New answers tagged asymptotics

1
vote

Accepted

### Lower bound on the number of balanced graphs

The bound of Ruciński and Vince is for strongly balanced, which is a more strict condition. If only balanced is required, the example of connected regular graphs provides a bound much greater than $n^{...

6
votes

Accepted

### Asymptotic behavior of a double oscillatory integral

$\newcommand{\R}{\mathbb R}\newcommand\sgn{\operatorname{sgn}}\newcommand{\vpi}{\varphi}$Obviously, for $a:=\sqrt{2\pi}\,\psi(0)$ we have
\begin{equation*}
\psi(0)=f(0),
\end{equation*}
where
\...

1
vote

### Calculate asymptotic value of an integral of exponential function

The strategy is as follows:
put everything in the integrand in the exponent $e^{f(z)}$, with
$f(z)=\zeta z-z^p/p$;
calculate the saddle point, the (possibly
complex) number $z_0$ where $f'(z)=0$; ...

1
vote

Accepted

### $\log(t)$ term in the small time expansion of $\mathrm{Tr}( A e^{-tB} )$

The answer by Carlo Beenakker is correct that any contribution to $D_n A_n \sim n^p$ with $p\ge 0$ does not generate any logarithmic asymptotic terms. Instead of numerical checks, one can also examine ...

0
votes

### $\log(t)$ term in the small time expansion of $\mathrm{Tr}( A e^{-tB} )$

I don't think there will be a logarithmic term. For definiteness, let me take $B_n=n^2$, $D_nA_n=n^p$, and approximate the sum over $n$ by an integral,
$$I_p(t)=\int_0^\infty n^p e^{-tn^2}=\tfrac{1}{2}...

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