I am reading the paper Duality of Quantum and Classical Error Correction Codes: Design Principles & Examples and at the beginning of it, the authors describe how the general Pauli channel is related to more realistic channels such as amplitude/phase damping channels. The Pauli channel is defined as:
$\mathcal{N}_P(\rho)=(1-p_x-p_y-p_z)I\rho I + p_xX\rho X+p_y Y\rho Y + p_z Z\rho Z$
In order to describe how such probabilities of $X,Y,Z$ events happen depending on the relaxation ($T_1$) and dephasing ($T_2$) times of a qubit, they give the next set of equations:
$p_x=p_y=\frac{1}{4}(1-e^{-\frac{t}{T_1}})$
$p_z=\frac{1}{4}(1+e^{-\frac{t}{T_1}}+ 2e^{-\frac{t}{T_2}})$
However, my doubt comes from the fact that intuitively, when the time goes to infinity, the probability of no error in the qubits should go to $0$, that is, complete decoherence of the system. However, making an analysis of this, this is not what happens:
$\lim_{t\rightarrow\infty} 1 - p_x-p_y-p_z = \lim_{t\rightarrow\infty}\frac{1}{4}(1+e^{-\frac{t}{T_1}}+2e^{-\frac{t}{T_2}})=\frac{1}{4}$.
So my question here is to clarify why such probability should go to $1/4$ instead of going to $0$, or in the case that someone knows more about this kind of relationships, to give some useful reference about it.