**Gadget simulations:**

We use the N-body code GADGET (Springel, 2005), which is a full MPI parallel code that uses the Particle-Mesh + Tree algorithms to compute the Newtonian forces between the dark matter particles by splitting the gravitational force into a long range term (computed through the PM method) and a short range one taken from the nearest neighbours, using a Tree method to categorize the particles according to their relative distances.

This code makes use of the public software library FFTW for parallel Fast-Fourier transforms and the GNU Scientific Library (GSL). We are using a non-public version of the code, named L-GADGET, which is highly optimized for large volume simulations with a cubic domain decomposition and an efficient use of internal memory. This code has been extensively used to produce large volume simulations with hundreds of billions of particles. The same code has been also used to produce other large simulations like the Multidark simulation suite and the Millenium series of simulations (including the Millennium XXL with more than 300 billion particles).

We use FastPM (Feng et al. 2016) to generate the paired initial conditions with fixed-amplitude.. The cosmological parameters for these simulations are taken from the Table 4 of Planck Collaboration (2016) with Ω_{m} = 0.3089, h ≡ H_{0}/100 = 0.6774, n_{s} = 0.9667 and σ_{8} = 0.8147. The box size is 1Gpc/h and the simulations start at a ≡ 1/(1 + z) = 0.01 (z = 99). The number of dark matter particles is 4096^{3} , which correspond to a particle mass of ∼ 1.2 × 10^{9} h^{−1}M_{sun}. We use the publicly available phase-.space halo finder code Rockstar (Behroozi et al. 2013) to identify haloes from the (129 ) snapshots we stored and then use the Consistent-Trees software to compute their merging histories.

**FastPM simulations:**

We use the C implementation of the FastPM software to perform the Particle-Mesh quasi-N-body simulations used in this project for the production of hundreds of dark matter paired simulations. Recent work showed that quasi-nbody simulations produced by COLA(Tassev et al. 2013), or FastPM(Feng et al. 2016) methods produce friends-of-friends halos that are very close to those from a full N-body simulation of identical mass resolution with much finer force and time resolution.

The FastPM code uses a pencil domain decomposition in the Poisson solver for gravity, and a Fourier space four-point differential kernel to compute the gravity force. The time integration scheme is modified from vanilla leap-frog to account for acceleration of velocity during a step and thus to correctly track the linear growth of large scale modes regardless of the number of time steps. The version used in this project does not contain a Friends-of-friends halo finder, therefore we use the finder provided by nbodykit (Hand et al. 2017) in a post-processing step to identify halos.

For both the 1Gpc/h and the 250Mpc/h boxes, the simulations start at a ≡ 1/(1 + z) = 0.01 (z = 99). A total of 100 timesteps were used to evolve the simulations to a = 1 (z = 0). We store the output boxes at z = 2, z = 1 and z = 0. When computing halos with the Friends-of-Friends halo finder in nbodykit, we chose a minimum of 20 dark matter particles per halo and a linking length of 0.2 Lbox/Nc. Here Lbox refers to the box size of the simulations and Nc is the number of cells used in the Particle-Mesh computation, which was taken to be the same as the number of dark matter particles.