Hi! I'm working on the ising model and i neede to solve two problems.
Let X_k from S={-1,1}^N to {-1,1} be the rv that assigns to each x in S his K-th componente x_k. Now consider the matrix Q(N) on S such that for x,y in S Q(N)_xy=1 if x differs from y by just one component (its k-th spin beign flipped), Q(N)_xx=-N, and Q(N)_xy=0 otherwise. Since its rows sums to 0 its esponential exp(tQ(N)) is a stochastic Matrix which represent a contnous time Markov process. Now fix N,t and let x_0 be the element consisting of all -1's. Consider the probability distribution P_N(y)=P(t,N)_x_0y, i.e the probability of going from x_0 to any other state.
Question: Show that under P_N the X_k's are independent identically distribuited (the second assertion is clear), i.e each component describe a markov process in S={-1,-1}.
I have no clue on how to do it, since its beyond human capability computing exactly exp(tQ(N)) (so i guess i need to make some different considerations...!!
ALso i think this must be due the nature of Q(N) because it wouldn't be true for any given Q-matrix....
Further question: Let A_N be the set such that (\sum_1^n X_k )/N is less or equal to -(e^(-2t))/2. Show that if L_N is the uniform probability on S and P_N is as above then P_N(A_N)-L_n(A_N) > 1-(8e^4t)/N. Note that the expectation of the X_k under P_N is -e(-2t) and is 0 under L_N.
Here one should use some Chebyshev estimate, i tried but i haven't succeeded.
Help on either one will be veery appreciated. Thanks in advance for any help, hint, solution, suggestion or whatever....
Let X_k from S={-1,1}^N to {-1,1} be the rv that assigns to each x in S his K-th componente x_k. Now consider the matrix Q(N) on S such that for x,y in S Q(N)_xy=1 if x differs from y by just one component (its k-th spin beign flipped), Q(N)_xx=-N, and Q(N)_xy=0 otherwise. Since its rows sums to 0 its esponential exp(tQ(N)) is a stochastic Matrix which represent a contnous time Markov process. Now fix N,t and let x_0 be the element consisting of all -1's. Consider the probability distribution P_N(y)=P(t,N)_x_0y, i.e the probability of going from x_0 to any other state.
Question: Show that under P_N the X_k's are independent identically distribuited (the second assertion is clear), i.e each component describe a markov process in S={-1,-1}.
I have no clue on how to do it, since its beyond human capability computing exactly exp(tQ(N)) (so i guess i need to make some different considerations...!!
ALso i think this must be due the nature of Q(N) because it wouldn't be true for any given Q-matrix....
Further question: Let A_N be the set such that (\sum_1^n X_k )/N is less or equal to -(e^(-2t))/2. Show that if L_N is the uniform probability on S and P_N is as above then P_N(A_N)-L_n(A_N) > 1-(8e^4t)/N. Note that the expectation of the X_k under P_N is -e(-2t) and is 0 under L_N.
Here one should use some Chebyshev estimate, i tried but i haven't succeeded.
Help on either one will be veery appreciated. Thanks in advance for any help, hint, solution, suggestion or whatever....