  Piecewise deterministic Markov processes are an important new tool in the
design of Markov Chain Monte Carlo algorithms. Two examples of fundamental
importance are the Bouncy Particle Sampler (BPS) and the Zig-Zag process (ZZ).
In this paper scaling limits for both algorithms are determined. Here the
dimensionality of the space tends towards infinity and the target distribution
is the multivariate standard normal distribution. For several quantities of
interest (angular momentum, first coordinate, and negative log-density) the
scaling limits show qualitatively very different and rich behaviour. Based on
these scaling limits the performance of the two algorithms in high dimensions
can be compared. Although for angular momentum both processes require only a
computational effort of $O(d)$ to obtain approximately independent samples, the
computational effort for negative log-density and first coordinate differ: for
these BPS requires $O(d^2)$ computational effort whereas ZZ requires $O(d)$.
Finally we provide a criterion for the choice of the refreshment rate of BPS.
