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the stationary MSD after many auctions have taken place. Consider again Equation (\ref{eq:gen-sol}) just before the $n+1$th auction at time $(n+1) \tau^-$, in the  case where the flow of new orders is symmetric, i.e. $\omega_{+}(y)=\omega_{-}(y)$, such that the transaction price is always at the fundamental price ($y^* = 0$). We will focus on  the supply side and postulate that $\rho_S(y,t=n \tau^-)$ can be written, in the vicinity of $y=0$, as \footnote{This approximation happens to be exact in the particular setting considered in \cite{donier2014fully}.}:  \be\label{eq:iteration} \begin{equation}\label{eq:iteration}  \rho_S(y,t=n \tau^-) = \sqrt{\tau} \, \phi_n\left(\frac{y}{\sqrt{\D \tau}}\right)+ O(\tau)  \ee  when $\tau \to 0$ (and symmetrically for the demand side). Plugging this ansatz into Equation (\ref{eq:gen-sol}), making the change of variable $y' \to \sqrt{\D \tau} w$   and taking the limit $\tau \to 0$ leads to the following iteration equation, exact up to order $\sqrt{\tau}$:\footnote{An extra correction of order $\sqrt{\tau}$ would appear if a drift term was added to Equation (\ref{eq:dynamics}).}  \be\nonumber \begin{equation}\nonumber  \phi_{n+1}(u) = \int_0^{+\infty} \frac{\d w}{\sqrt{4 \pi}} \phi_n(w) e^{-(u-w)^2/4} + \sqrt{\tau} \, \omega(0) + O(\tau).  \ee  Note that $\nu$ has entirely disappeared from the equation (but will appear in the boundary condition, see below), and only the value of $\omega$ close to the transaction price is relevant at this order.  After a very large number of auctions, one therefore finds that the stationary shape of the demand curve close to the price and in the limit $\tau \to 0$ is given by the non-trivial solution of the following   fixed point equation:  \be\label{eq:atkinson} \begin{equation}\label{eq:atkinson}  \phi_\infty(u) = \int_0^{+\infty} \frac{\d w}{\sqrt{4 \pi}} \phi_\infty(w) e^{-(u-w)^2/4},  \ee  supplemented by the boundary condition $\phi_\infty(u \gg 1) \approx \L \sqrt{\D} u$, where $\L$ is a constant to be determined below. [Note that the solution of Equation (\ref{eq:atkinson}) is determined up to a   multiplicative factor that must be fixed by some external condition].