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The present paper is concerned with properties of multiple Schramm--Loewnerevolutions (SLEs) labelled by a parameter $\kappa\in (0,8]$. Specifically, weconsider the solution of the multiple Loewner equation driven by a time changeof Dyson's Brownian motions in the non-colliding regime. Although it is oftenconsidered that several properties of the solution can be studied by means ofcommutation relations of SLEs and the absolute continuity, this method isavailable only in the case that the curves generated by commuting SLEs areseparated. Beyond this restriction, it is not even obvious that the solution ofthe multiple Loewner equation generates multiple curves. To overcome thisdifficulty, we employ the coupling of Gaussian free fields and multiple SLEs.Consequently, we prove the longstanding conjecture that the solution indeedgenerates multiple continuous curves. Furthermore, these multiple curves are(i) simple disjoint curves when $\kappa\in (0,4]$, (ii) intersecting curveswhen $\kappa\in (4,8)$, and (iii) space-filling curves when $\kappa=8$.

Three phases of multiple SLE driven by non-colliding Dyson's Brownian motions