Large $N$ limit and $1/N$ expansion of invariant observables in $O(N)$ linear $\sigma$-model via SPDE
A talk in the Bielefeld Stochastic Afternoon series by
Rongchan Zhu
| Abstract: | In this talk, I will talk about the study of large $N$ problems for the
Wick renormalized linear sigma model, i.e. $N$-component $\Phi^4$
model, in two spatial dimensions, using stochastic quantization methods
and Dyson--Schwinger equations. We identify the large $N$ limiting law
of a collection of Wick renormalized $O(N)$ invariant observables. In
particular, under a suitable scaling, the quadratic observables converge
in the large $N$ limit to a mean-zero (singular) Gaussian field denoted
by $\cQ$ with an explicit covariance;
and the observables which are renormalized powers of order $2n$
converge in the large $N$ limit to suitably renormalized $n$-th powers
of $\mathcal{Q}$.
The quartic interaction term of the model has no effect on the large $N$
limit of the field, but has nontrivial contributions to the limiting law
of the observables, and the renormalization of the $n$-th powers of
$\mathcal{Q}$ in the limit has an interesting finite shift from the
standard one.
Furthermore, we derive the $1/N$ asymtotic expansion for the $k$-point
functions of the quadratic observables by employing graph
representations and analyzing the order of each graph from
Dyson--Schwinger equations. Finally, turning to the stationary solutions
to the stochastic quantization equations, with the Ornstein--Uhlenbeck
process being the large $N$ limit, we derive here its next order
correction in stationarity, as described by an SPDE with the right-hand
side having explicit marginal law which involves the above field
$\mathcal{Q}$. Within the CRC this talk is associated to the project(s): B1 |