Geometric Convergence p):distribution at time twhen initial state isx △(t)=lp-πlTv△(t)=max△x(t) x∈2 Tx(e)=min{t|△x(t)≤e}r(e)=max Ta(e) x∈2 Tmix =T(1/2e) Theorem:△(k,Tmix)≤ek 三X22%w-g9vse x =X1,X2,... 3 coupling Pr[Xt≠YAl≤e-1 couple X=Y,→X+1=Y+l in caseX=r'≠y'=y子coupling P≠Yl=lpg-t-pg-tIlrGeometric Convergence ￾x(t) = kp(t) x ￾ ⇡kT V p(t) x : distribution at time t when initial state is x Theorem: ￾(k · ⌧mix)  e￾k ￾(t) = max x2⌦ ￾x(t) ⌧x(✏) = min{t | ￾x(t)  ✏} ⌧ (✏) = max x2⌦ ⌧x(✏) ⌧mix = ⌧ (1/2e) x = X1, X2, ..., Xt, Xt+1, ... , Xkt, ... y = Y1, Y2, ..., Yt, Yt+1, ... , Ykt, ... kp(t) x ￾ p(t) y kT V  e￾1 Pr[Xt 6= Yt]  e ∃ coupling ￾1 couple Xt=Yt ⇒ Xt+1=Yt+1 in case Xt=x’ ≠ y’=Yt ∃ coupling Pr[Xkt 6= Ykt] = kp (k￾1)t x0 ￾ p (k￾1)t y0 kT V