English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider the 0-dimensional quartic $O(N)$ vector model and present acomplete study of the partition function $Z(g,N)$ and its logarithm, the freeenergy $W(g,N)$, seen as functions of the coupling $g$ on a Riemann surface.Using constructive field theory techniques we prove that both $Z(g,N)$ and$W(g,N)$ are Borel summable functions along all the rays in the cut complexplane $\mathbb{C}_{\pi} =\mathbb{C}\setminus \mathbb{R}_-$. We recover thetransseries expansion of $Z(g,N)$ using the intermediate field representation.We furthermore study the small-$N$ expansions of $Z(g,N)$ and $ W(g,N)$. Forany $g=|g| e^{\imath \varphi}$ on the sector of the Riemann surface with$|\varphi|<3\pi/2$, the small-$N$ expansion of $Z(g,N)$ has infinite radius ofconvergence in $N$ while the expansion of $W(g,N)$ has a finite radius ofconvergence in $N$ for $g$ in a subdomain of the same sector. The Taylorcoefficients of these expansions, $Z_n(g)$ and $W_n(g)$, exhibit analyticproperties similar to $Z(g,N)$ and $W(g,N)$ and have transseries expansions.The transseries expansion of $Z_n(g)$ is readily accessible: much like$Z(g,N)$, for any $n$, $Z_n(g)$ has a zero- and a one-instanton contribution.The transseries of $W_n(g)$ is obtained using M\"oebius inversion and summingthese transseries yields the transseries expansion of $W(g,N)$. The transseriesof $W_n(g)$ and $W(g,N)$ are markedly different: while $W(g,N)$ displayscontributions from arbitrarily many multi-instantons, $W_n(g)$ exhibitscontributions of only up to $n$-instanton sectors.

The small-$N$ series in the zero-dimensional $O(N)$ model: constructive expansions and transseries