## Rachunek Prawdopodobieństwa 2

### Zestaw zadań nr 8

### Termin realizacji: 16 I 2008

### 1. Show that the Markov chain used for the MCMC simulation of the hard-core model (Example 1) is

*• irreducible*

*• aperiodic.*

*2. Show that the Gibbs sampler for random q-colorings (Example 2)*

*• has ρ*

*G,q*

### as a stationary distribution,

*• is aperiodic, and,*

*• for q ≥ ∆(G) + 2, it is irreducible.*

*3. Let G = (V, E) be a connected graph and let a function X be chosen uniformly at random* *from [q]*

^{V}*. Show that the probability that X is a q-coloring is at most*

### µ *q − 1* *q*

### ¶

_{k−1}*.*

*4. Udowodnić, że jeśli 0 ≤ ² ≤ 1, a*

1*, . . . , a*

*k*

*, b*

1*, . . . , b*

*k*

*> 0 i, dla wszystkich j = 1, . . . , k*

*1 −* *²* *2k* *≤* *a*

*j*

*b*

*j*

*≤ 1 +* *²* *2k* *,* to

*1 − ² ≤* Q

_{k}*j=1*

*a*

*j*

### Q

_{k}*j=1*

*b*

*j*

*≤ 1 + ².*

*5. (Generalized hard-core model) Given a graph and a parameter λ > 0, define for each configu-* *ration ξ ∈ {0, 1}*

^{V}### a probability measure

*µ*

*G,λ*

*(ξ) =* *λ*

^{n(ξ)}*Z*

*G,λ*

*if ξ is feasible and 0 otherwise, where n(ξ) is the number of 1’s in ξ and*

*Z*

*G,λ*

### = X

*(ξ*

### feasible

)*λ*

^{n(ξ)}### is a normalizing constant.

*(a) Compute the conditional probability that a vertex v ∈ V takes the value 1, given that all* *neighbors of v take value 0.*

### (b) Construct an MCMC algorithm for this generalized hard-core model.

*6. The error probability 1/3 in an rptas can be cut down to any given δ > 0 by the following* *method: Run the algorithm many times (say m, where m is odd), and take the median of the* outputs (i.e., the

^{m+1}_{2}