R_\mathrm{vir}}},\\
&\approx & 275~{\rm km}\,{\rm s}^{-1} \left (\frac{M_{\rm halo}}{10^{12}~{\rm M}_\odot}\right )^{1/3} \left (\frac{1+z}{3}\right )^{1/2}, \end{aligned}$$ where we assumed a matter-dominated Universe and used $$\begin{aligned}
R_\mathrm{vir} &=& \left(\dfrac{2GM}{H_0^2\Omega_\mathrm{m}\Delta_c}\right)^{1/3}\dfrac{1}{1+z},\\
&\approx & 114~{\rm kpc} \left( \frac{M_{\rm halo}}{10^{12}~{\rm M}_\odot}\right )^{1/3} \left (\frac{1+z}{3}\right )^{-1}.\end{aligned}$$ We show $-v_\mathrm{esc}$ by the dashed, grey curve. We only expect the gas to have a velocity close to this estimate if it fell in freely from very large distances and if the Hubble expansion, which damps peculiar velocities, were unimportant. However, we do expect the scaling with mass to be more generally applicable. For the cold mode the trend with halo mass is indeed well reproduced by the escape velocity.

![image](figures/radialpropflux_REF_L050N512_z2p0_mass11p5to12p5_pecvel0p1.eps)

![image](figures/masspropflux_REF_L050N512_z2p0_pecvel0p1.eps)

On average, for halo masses above $10^{11}$ M$_\odot$, the net accretion rate is positive, which means that more mass is flowing in than is flowing out (bottom-middle panel). Therefore, unsurprisingly, the haloes are growing. For haloes with $10^{10}$ M$_\odot<M_\mathrm{halo}<10^{11}$ M$_\odot$ the mean accretion rate is negative (indicating net outflow), but small ($\sim 0.1$ M$_\odot \,$yr$^{-1}$). The mean accretion rate of cold-mode gas is positive for all halo masses, but for hot-mode gas there is net outflow for $M_\mathrm{halo}<10^{11.5}$ M$_\odot$. Although these haloes are losing gas that is currently hot-mode, their hot-mode gas reservoir may still be increasing if cold-mode gas is converted into hot-mode gas. For higher-mass haloes, the hot-mode accretion rate is also positive and it increases approximately linearly with halo mass. This is the regime where the implemented supernova feedback is not strong enough to blow gas out of the halo. This transition mass is increased