  The superiority of stochastic symplectic methods over non-symplectic
counterparts has been verified by plenty of numerical experiments, especially
in capturing the asymptotic behaviour of the underlying solution process. How
can one theoretically explain this superiority? This paper gives an answer to
this problem from the perspective of the law of iterated logarithm, taking the
linear stochastic Hamiltonian system in Hilbert space as a test model. The main
contribution is twofold. First, by fully utilizing the time-change theorem for
martingales and the Borell--TIS inequality, we prove that the upper limit of
the exact solution with a specific scaling function almost surely equals some
non-zero constant, thus confirming the validity of the law of iterated
logarithm. Second, we prove that stochastic symplectic fully discrete methods
asymptotically preserve the law of iterated logarithm, but non-symplectic ones
do not. This reveals the good ability of stochastic symplectic methods in
characterizing the almost sure asymptotic growth of the utmost fluctuation of
the underlying solution process. Applications of our results to the linear
stochastic oscillator and the linear stochastic Schrodinger equation are also
presented.
