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regularity is at the heart of position space ( block spin ) renormalization group .
it is usually assumed , and of central importance , that coupling constants of the block spin effective hamiltonian depend in a non - singular way on the parameters of the original theory .
there are , however , situations where this assumption is not valid .
pathological behaviour in renormalization groups of the low temperature ising model was first observed by israel @xcite and griffith and pearce @xcite .
an extensive and rigorous analysis of regularity properties and pathologies in ising model block spin transformations was performed by van enter et al .
the central observation is that in certain situations the effective measure for the block spin theory can not be represented as @xmath2 .
this means that the effective measure is non - gibbsian .
see also @xcite for a careful analysis of the situation .
in this paper , i present some numerical results on the distribution of zeroes in the complex @xmath1 plane for block spin boltzmannians of the 2d ising model .
these boltzmannians are partition functions with `` fixed '' block spins @xmath3 , viz .
b ( ) = _ p ( , ) .
@xmath4 encodes the blocking rule .
why should these zeroes provide interesting information ?
if the usual renormalization group assumptions are true , the zeroes of @xmath5 should  for all block spin configurations @xmath3  behave `` better '' than those of the full partition function . note that the zeroes of the full partition function approach the real axis at the critical point @xcite .
this should _ not _ happen for the zeroes of @xmath5 !
furthermore , one might expect that the pathologies described in the literature are related to the distribution of zeroes close to the @xmath1 axis at large positive values .
this article is organized as follows : in section 2 the model notation is set up , and the blocking rules are defined .
section 3 gives a sketch of the transfer matrix technique used to compute the polynomials .
section 4 summarizes the numerical results for the majority rule blocking .
observations for the kadanoff blocking rule are reported in section 5 .
conclusions follow .
we deal with the 2d ising model , with partition function z = _ , where h()= - _ <
i , j > _ i _ j . the @xmath6 assume values @xmath7 and are defined on a square lattice of extension @xmath8 by @xmath8 , supplied with periodic boundary conditions .
the energy @xmath9 is a sum over all pairs of nearest neighbours . in the infinite volume limit
the model undergoes a second order phase transition at @xmath10 .
a block spin transformation with scale factor 2 is defined as follows . for @xmath8
even , the lattice is divided in blocks of size 2 by 2 .
given the configuration of the @xmath11-spins in a block @xmath12 , a block spin @xmath13 is chosen with probability @xmath4 .
the majority rule is defined through p ( , ) = _ blocks i p_i(_i , ) , with p_i(_i , ) = \ {    ll 12 & _ ii _ i = 0 , + 1 & _ i _ ii _
i > 0 , + 0 & .    .
the so - called kadanoff rule is [ kadanoff ] p_i(_i , ) = . in the limit @xmath14
one recovers the majority rule .
given that @xmath15 , the full partition function can be rewritten as z = _ b ( ) , with b ( ) = _ p ( , ) .
usually , one aims at a parametrisation @xmath16 $ ] , where @xmath17 is the effective hamiltonian .
note , however , that this is impossible in the pathological situations discussed in the literature @xcite .
an exact computation of @xmath5 as function of @xmath1 seems impossible . however , on lattices up to at least @xmath18 it can be determined by numerical transfer matrix calculations .
@xmath18 corresponds to 4 by 4 blocks , with @xmath19 block spin configurations @xmath3 .
we do not have to consider them all .
configurations connected through global spin flip or geometric symmetries ( reflections , shifts , rotations ) have the same effective boltzmannian @xmath5 . a careful counting yields for the number of non - equivalent configurations @xmath20    [ cols="^,^,^,^,^",options="header " , ]     in figures [ fig5 ] and [ fig6 ] we show the zeroes for @xmath21 and @xmath22 , both for @xmath23 and @xmath24 .
obviously , most of the zeroes in the neighbourhood of the critical circle do not move very much .
in fact they are already very close to their majority rule values .
however , compared with the @xmath25 case , extra zeroes appear that populate the region of larger real part of @xmath1
. it might be interesting to note that the zeroes with the largest real part of @xmath1 belong to block spin configuration @xmath26 , followed by those of @xmath27 .
another observation , also clearly seen in figure [ fig6 ] , is that the spurious zeroes move to the right when @xmath28 is increased .
most likely they are shifted to infinity when passing to the majority rule .                in figure 7
the zeroes of all three lattice sizes , @xmath21 , 6 , and 8 are plotted for the case @xmath29 . , some 16 of the 479 block spin configurations were not taken into account because the search for the zeroes of the corresponding polynomials suffered from numerical instabilities . ]
a careful inspection reveals that the zeroes in the right half plane _ do _ move towards the real axis when the lattice size is increased .
they might thus very well be reflecting the large @xmath1 pathologies .
the distribution of zeroes of effective boltzmannians was studied for 2d ising systems .
both in the case of the majority and the kadanoff rule a finite region around the critical point stays free of zeroes , also when the volume is increased . in case of the kadanoff rule , however , zeroes populate the right half plane and approach the real axis at large @xmath1 .
they might be related to the pathologies of a number of ising renormalization groups discussed in the literature .
it would be very interesting to understand this relation ( if it exists ) .
furthermore , one should try to understand the origin of the extra zeroes that appear for finite @xmath28 .
van enter et al .
were not able to extend their analysis of pathologies to @xmath25 in two dimensions .
the present study shows that the distribution of large @xmath1 zeroes changes significantly when @xmath28 becomes finite .
the present findings do not , however , exclude the possibility of large @xmath1 pathologies of the 2d ising majority rule renormalization group .
effective boltzmannians in the sense of the block spin renormalization group are computed for the 2d ising model . 
 the blocking is done with majority and kadanoff rules for blocks of size 2 by 2 . 
 transfer matrix techniques allow the determination of the effective boltzmannians as polynomials in @xmath0 for lattices of up to 4 by 4 blocks . 
 the zeroes of these polynomials are computed for all non - equivalent block spin configurations . 
 their distribution in the complex @xmath1 plane reflects the regularity structure of the block spin transformation . in the case of 
 the kadanoff rule spurious zeroes approach the positive real @xmath1 axis at large values of @xmath1 . 
 they might be related to the renormalization group pathologies discussed in the literature .    
 0.5 cm    ms  
 tpi9815    0.8 cm    0.8 cm    klaus pinn +   + universitt mnster   + wilhelm  klemm  str 
 .  9   + d48149 mnster , germany + e  mail : pinn@unimuenster.de    2.0 cm