What is Arbitrary?

Legal Definition
What depends on the will of the judge, not regulated or established by law. Bacon (Aphor. 8) says, Optima lex quae minimum relinquit arbitrio judicis et (Aph. 46) optimus judex, qui mi nimum sibi

2. In all well adjusted systems of law every thing is regulated, and nothing arbitrary can be allowed; but there is a discretion which is sometimes allowed by law which leaves the judge free to act as he pleases to a certain extent. See Discretion
-- Bouviers Law Dictionary
Legal Definition
Not supported by fair, solid, and substantial cause, and without reason given. Treloar v. Bigge, L. R. 9 Exch. 155.
See also
-- Black's Law Dictionary
Legal Definition
That which is decided by the judge on his own judgment and not under a statute.
-- Ballentine's Law Dictionary