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Erratum to: Return Probabilities for the Reflected Random Walk on \(\mathbb {N}_0\)
Erratum to: Return Probabilities for the Reflected Random Walk on N0
Rim Essifi
Marc Peign
In the original publication of this paper, we fix a constant K > 1 and consider the set K(K) of functions K : Z R+ satisfying the following conditions: Unfortunately, the two first conditions readily imply K = 1 since the operator R is markovian, so that the three above conditions cannot be satisfied simultaneously. In fact, we will simply consider the function K : s Kx . The only one reason for the condition RK (x ) 1 appeared in the proof of Fact 4.4.1, where the peripherical spectrum of the operators Rs for s = 1 and s = 1 is controlled. With this new choice of function K , one gets rFaadcitus4.o4f.1RFsoorns(C=0N, 1 anKd )sis=<11o.ne gets Rs K < 1; in particular, the spectral Proof We could adapt the proof proposed in the paper and show that Rs2n for some s < 1 when s = 1. We propose here another simpler argument.

x N0
K (x ) 1,
RK (x ) 1 and
K (x ) Kx .
comRpeaccatllfrthoamt R(CsNa,ctsfro)min(toC(NC,N K ) into (CN ,  ) and that the identity map is
,  K ). Consequently, the operator RK is compact
on (CN ,  K ) with spectral radius 1 since it has bounded powers.
Let us fix s C\{1} with modulus 1 and assume that Rs has spectral radius 1 on
(CN ,  K ); since it is compact, there exists a sequence a = (ax )xZ = 0 and R
such that Rs a = ei a, i.e.
for all x Z :
yZ
for all x Z :
yZ
Acknowledgments We thank here S. Gouezel, who pointed out to us the contradiction of the three
conditions (1) and the fact that the Rs are compact on (C,  K ).