In this talk, Amir Dembo considers random walks on Z^{2} and presents a proof of the ErdösTaylor conjecture related to frequently covered points. The KestenRévész conjecture on the covering time of the twodimensional torus Z_{n}^{2}=Z^{2}/nZ^{2} is also solved. These results are a common work of Amir Dembo, Yuval Peres, Jay Rosen, and Ofer Zeitouni.


= 

, almost surely. (1) 
T  _{n}= 

T(x), 


= 

, in probability. (2) 

® 0, or equivalently, 

® 1 (as n®¥), (3) 
P(X>t)£ 

E (X^{2}). 
P(X=0)£ P  (  Xµ³µ  )  £ 

, for µ=E X. 
G_{n}(0,0)= 

E  (  1_{{Xj=0}}  )  = 

P(X_{j}=0)~ 

. 
P  (  X_{j}¹ 0 for j=1,...,n1  )  £ 

. 
P  (  T_{n}(0)³ ap(logn)^{2}  )  £ 
æ ç ç è 
1 

ö ÷ ÷ ø 
^{a(logn)2}£ e^{ap(logn)(1d)}= n^{(1d)ap}. (5) 
Z_{n}= 

1 

. 



T_{m}^{*}= 

T_{m}(x), 
H_{y}=H_{y}(u)= 

P^{y}  (  X spends time k at a before hitting 0  )  u^{k}. 
H_{1}= 

+ 

+ 

, H_{k}= 

+ 

+ 

(2£ k £ m2), H_{m1}= 

+ 

. 
P  (  T_{m}(a)>a m^{2}  )  = 
æ ç ç è 
æ ç ç è 
1 

ö ÷ ÷ ø 
^{m} 
ö ÷ ÷ ø 
^{a m}~ e^{a m} and P(T_{m}^{*}>a m^{2})£ e^{a m} 2^{m}=e^{(alog2)m}. 
N_{i}(x)~ a i^{2}, for i=0, K, 2K, ...,K 
ê ê ê ë 

ú ú ú û 
. 
P  (  N_{i+K}(x)~a(i+K)^{2}  N_{i}(x)~a i^{2}  )  ~ e^{a K} Þ P(xray is asuccessfull)~ e^{a m}. 
Z_{m}= 

1_{{xray asuccessfull}}, 

~ 

e^{(alog 2)Ks}® 1 for a<log2, 
q=min  {  t  w_{t}=1  } 
and  µ_{q}^{w}(A)=  ó õ 

1_{A}(w_{t})dt. 

P 
æ ç ç è 

½ ½ ½ ½ 
w_{k/n} 

S_{k} 
½ ½ ½ ½ 
> d n^{h1/2} 
ö ÷ ÷ ø 
=0. (9) 

n(1+2d)^{3}e_{n}^{2}. 
P^{a}(X hits x before 0)=1H_{1}(0)= 

. 
P^{0}(X does not cover x during first N visits to 0) ~ 
æ ç ç è 
1 

ö ÷ ÷ ø 
^{N}. 
P^{0}£2^{m} 
æ ç ç è 
1 

ö ÷ ÷ ø 
^{N} so that P^{0}® 0 for N=2(1+d)m^{2}log 2. 
P^{0}  (  X does not cover B_{m} before time 2(1+d)log2× m^{2} 2^{m+1}  )  ® 0. 
T(x,e)=inf  {  t>0  w_{t}Î D(x,e)  }  , and  C_{e}= 

T(x,e). 


= 

, almost surely. 
This document was translated from L^{A}T_{E}X by H^{E}V^{E}A.