Let (r_n) be a positive nondecreasing sequence of finite genus tending to +∞ , and (η_n(ω)) be a sequence of independent random variables such that η_n(ω) are uniformly distributed on the circles |z|=r_n. Then for almost all ω the following assertion holds: if f is an entire function of finite order with zeros at the points η_n(ω) and only at them, then for every ε>0 we have ln M_f(r)=o(T^3/2_f(r)ln^{3+ε}T_f(r)), r→+∞, where M_f(r) is the maximum modulus and T_f(r) is the Nevanlinna characteristic of the function f.