note on basic features of large time behaviour of heat kernels large time behaviour of heat semigroups and more gener ally of positive selfadjoint semigroups is studied convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoint semigroups this framework includes laplacians on manifolds metric graphs and discrete graphs

exact t 0 partition functions for potts antiferromagnets on sections of the simple cubic lattice we present exact solutions for the zero temperature partition function of the q state potts antiferromagnet equivalently the chromatic polynomial p on tube sections of the simple cubic lattice of fixed transverse size l x ifmmode times else texttimes fi l y and arbitrarily great length l z for sizes l x ifmmode times else texttimes fi l y 2 ifmmode times else texttimes fi 3 and 2 ifmmode times else texttimes fi 4 and boundary conditions a mathrm fbc x mathrm fbc y mathrm fbc z and b mathrm pbc x mathrm fbc y mathrm fbc z where fbc pbc denote free periodic boundary conditions in the limit of infinite length l z ensuremath rightarrow ensuremath infty we calculate the resultant ground state degeneracy per site w exponent of the ground state entropy generalizing q from z to c we determine the analytic structure of w and the related singular locus mathcal b which is the continuous accumulation set of zeros of the chromatic polynomial for the l z ensuremath rightarrow ensuremath infty limit of a given family of lattice sections w is analytic for real q down to a value q c we determine the values of q c for the lattice sections considered and address the question of the value of q c for a d dimensional cartesian lattice analogous results are presented for a tube of arbitrarily great length whose transverse cross section is formed from the complete bipartite graph k m m

complex temperature singularities in potts models on the square lattice we report some results on the complex temperature ct singularities of q state potts models on the square lattice we concentrate on the problematic region mathrm re a l0 where a e k in which ct zeros of the partition function are sensitive to finite lattice artifacts from analyses of low temperature series expansions for 3l ql 8 we establish the existence in this region of complex conjugate ct singularities at which the magnetization and susceptibility diverge from calculations of zeros of the partition function we obtain evidence consistent with the inference that these singularities occur at endpoints a e a e of arcs protruding into the complex temperature extension of the ferromagnetic phase exponents for these singularities are determined e g for q 3 we find ensuremath beta e ensuremath 0 125 1 consistent with ensuremath beta e ensuremath frac 1 8 by duality these results also imply associated arcs extending into the ct extension of the symmetric paramagnetic phase analytic expressions are suggested for the positions of some of these singularities e g for q 5 our finding is consistent with the exact value a e a e 2 ensuremath 1 ensuremath mp i further discussions of complex temperature phase diagrams are given

exact partition functions for potts antiferromagnets on cyclic lattice strips abstract we present exact calculations of the zero temperature partition function of the q state potts antiferromagnet on arbitrarily long strips of the square triangular and kagome lattices with width l y 2 or 3 vertices and with periodic longitudinal boundary conditions from these in the limit of infinite length we obtain the exact ground state entropy s 0 k b ln w these results are of interest since this model exhibits nonzero ground state entropy s 0 0 for sufficiently large q and hence is an exception to the third law of thermodynamics we also include results for homeomorphic expansions of the square lattice strip the analytic properties of w q are determined and related to zeros of the chromatic polynomial for long finite strips

ground state entropy of the potts antiferromagnet on strips of the square lattice abstract we present exact solutions for the zero temperature partition function chromatic polynomial p and the ground state degeneracy per site w exponent of the ground state entropy for the q state potts antiferromagnet on strips of the square lattice of width ly vertices and arbitrarily great length lx vertices the specific solutions are for a ly 4 fbcy pbcx cyclic b ly 4 fbcy tpbcx mobius c ly 5 6 pbcy fbcx cylindrical and d l y 5 fbc y fbc x open where fbc pbc and tpbc denote free periodic and twisted periodic boundary conditions respectively in the lx limit of each strip we discuss the analytic structure of w in the complex q plane the respective w functions are evaluated numerically for various values of q several inferences are presented for the chromatic polynomials and analytic structure of w for lattice strips with arbitrarily great ly the absence of a nonpathological lx limit for real nonintegral q in the interval 0 0 for strips of the square triangular lattice is discussed

volume growth spectrum and stochastic completeness of infinite graphs we study the connections between volume growth spectral properties and stochastic completeness of locally finite weighted graphs for a class of graphs with a very weak spherical symmetry we give a condition which implies both stochastic incompleteness and discreteness of the spectrum we then use these graphs to give some comparison results for both stochastic completeness and estimates on the bottom of the spectrum for general locally finite weighted graphs

complex temperature phase diagrams for the q state potts model on self dual families of graphs and the nature of the q limit exact calculations of the potts model partition function z g q v have been presented for arbitrary q and temperature like variable v on self dual strip graphs g of the square lattice with fixed width l y and arbitrarily great length lx with two types of boundary conditions letting lx infinity the resultant free energy and complex temperature phase diagram have been computed including the locus b where the free energy is nonanalytic results are analyzed for widths l y 1 2 3 these results have been used to study the approach to the large q limit of b

transfer matrices and partition function zeros for antiferromagnetic potts models iii triangular lattice chromatic polynomial we study the chromatic polynomial pg q for m n triangular lattice strips of widths m 12p 9f with periodic or free transverse boundary conditions respectively and arbitrary lengths n with free longitudinal boundary conditions the chromatic polynomial gives the zero temperature limit of the partition function for the q state potts antiferromagnet we compute the transfer matrix for such strips in the fortuin kasteleyn representation and obtain the corresponding accumulation sets of chromatic zeros in the complex q plane in the limit n we recompute the limiting curve obtained by baxter in the thermodynamic limit m n and find new interesting features with possible physical consequences finally we analyze the isolated limiting points and their relation with the beraha numbers

chromatic polynomials for j h i strip graphs and their asymptotic limits abstract we calculate the chromatic polynomials p for n vertex strip graphs of the form j l 1 m h i where j and i are various subgraphs on the left and right ends of the strip whose bulk is comprised of m fold repetitions of a subgraph h the strips have free boundary conditions in the longitudinal direction and free or periodic boundary conditions in the transverse direction this extends our earlier calculations for strip graphs of the form l 1 m h i we use a generating function method from these results we compute the asymptotic limiting function w lim n p 1 n for q z this has physical significance as the ground state degeneracy per site exponent of the ground state entropy of the q state potts antiferromagnet on the given strip in the complex q plane w is an analytic function except on a certain continuous locus b in contrast to the l 1 m h i strip graphs where b i is independent of i and ii consists of arcs and possible line segments that do not enclose any regions in the q plane we find that for some j l 1 m h i strip graphs b i does depend on i and j and ii can enclose regions in the q plane our study elucidates the effects of different end subgraphs i and j and of boundary conditions on the infinite length limit of the strip graphs

asymptotic limits and zeros of chromatic polynomials and ground state entropy of potts antiferromagnets we study the asymptotic limiting function w g q lim n to infty p g q 1 n where p g q is the chromatic polynomial for a graph g with n vertices we first discuss a subtlety in the definition of w g q resulting from the fact that at certain special points q s the following limits do not commute lim n to infty lim q to q s p g q 1 n ne lim q to q s lim n to infty p g q 1 n we then present exact calculations of w g q and determine the corresponding analytic structure in the complex q plane for a number of families of graphs g including circuits wheels biwheels bipyramids and cyclic and twisted ladders we study the zeros of the corresponding chromatic polynomials and prove a theorem that for certain families of graphs all but a finite number of the zeros lie exactly on a unit circle whose position depends on the family using the connection of p g q with the zero temperature potts antiferromagnet we derive a theorem concerning the maximal finite real point of non analyticity in w g q denoted q c and apply this theorem to deduce that q c sq 3 and q c hc 3 sqrt 5 2 for the square and honeycomb lattices finally numerical calculations of w hc q and w sq q are presented and compared with series expansions and bounds

exact potts model partition functions on strips of the honeycomb lattice abstract we present exact calculations of the partition function of the q state potts model on i open ii cyclic and iii mobius strips of the honeycomb brick lattice of width l y 2 and arbitrarily great length in the infinite length limit the thermodynamic properties are discussed the continuous locus of singularities of the free energy is determined in the q plane for fixed temperature and in the complex temperature plane for fixed q values we also give exact calculations of the zero temperature partition function chromatic polynomial and w q the exponent of the ground state entropy for the potts antiferromagnet for honeycomb strips of type iv l y 3 cyclic v l y 3 mobius vi l y 4 cylindrical and vii l y 4 open in the infinite length limit we calculate w q and determine the continuous locus of points where it is nonanalytic we show that our exact calculation of the entropy for the l y 4 strip with cylindrical boundary conditions provides an extremely accurate approximation to a few parts in 10 5 for moderate q values to the entropy for the full 2d honeycomb lattice where the latter is determined by monte carlo measurements since no exact analytic form is known

the potts model and flows ii many spin correlation function for pt i see ibid vol 19 p 411 1985 a graph theoretic analysis is made of the m spin correlation functions of the lambda state potts model in paper i the correlation functions for m 2 were expressed in terms of rooted mod lambda flow polynomials the authors introduce a more general type of polynomial the partitioned m rooted flow polynomial which plays a fundamental role in the calculation of the multispin correlation functions the m rooted equivalent transmissivities of tsallis and levy 1981 are interpreted in terms of percolation theory and are expressed as linear combinations of the above correlation functions

analytic and geometric background of recurrence and non explosion of the brownian motion on riemannian manifolds we provide an overview of such properties of the brownian motion on complete non compact riemannian manifolds as recurrence and non explosion it is shown that both properties have various analytic characterizations in terms of the heat kernel green function liouville properties etc on the other hand we consider a number of geometric conditions such as the volume growth isoperimetric inequalities curvature bounds etc which are related to recurrence and non explosion

on minimal parabolic functions and time homogeneous parabolic h transforms burdzy was supported in part by nsf grant dms 9700721 saliosbury was supported in part by a grant from nserc a portion of this research took place during his stay at the fields institute

general structural results for potts model partition functions on lattice strips abstract we present a set of general results on structural features of the q state potts model partition function z g q v for arbitrary q and temperature boltzmann variable v for various lattice strips of arbitrarily great width ly vertices and length lx vertices including i cyclic and mobius strips of the square and triangular lattices and ii self dual cyclic strips of the square lattice we also present an exact solution for the chromatic polynomial for the cyclic and mobius strips of the square lattice with width ly 5 the greatest width for which an exact solution has been obtained so far for these families in the lx limit we calculate the ground state degeneracy per site w q and determine the boundary b across which w q is singular in the complex q plane

exact potts model partition functions on wider arbitrary length strips of the square lattice abstract we present exact calculations of the partition function of the q state potts model for general q and temperature on strips of the square lattice of width l y 3 vertices and arbitrary length l x with periodic longitudinal boundary conditions of the following types i fbc y pbc x cyclic ii fbc y tpbc x mobius iii pbc y pbc x toroidal and iv pbc y tpbc x klein bottle where fbc and t pbc refer to free and twisted periodic boundary conditions results for the l y 2 torus and klein bottle strips are also included in the infinite length limit the thermodynamic properties are discussed and some general results are given for low temperature behavior on strips of arbitrarily great width we determine the submanifold in the c 2 space of q and temperature where the free energy is singular for these strips our calculations are also used to compute certain quantities of graph theoretic interest

families of graphs with w r g q functions that are nonanalytic at 1 q 0 denoting p g q as the chromatic polynomial for coloring an n vertex graph g with q colors and considering the limiting function w g q lim n to infty p g q 1 n a fundamental question in graph theory is the following is w r g q q 1 w g q analytic or not at the origin of the 1 q plane where the complex generalization of q is assumed this question is also relevant in statistical mechanics because w g q exp s 0 k b where s 0 is the ground state entropy of the q state potts antiferromagnet on the lattice graph g and the analyticity of w r g q at 1 q 0 is necessary for the large q series expansions of w r g q although w r is analytic at 1 q 0 for many g there are some g for which it is not for these w r has no large q series expansion it is important to understand the reason for this nonanalyticity here we give a general condition that determines whether or not a particular w r g q is analytic at 1 q 0 and explains the nonanalyticity where it occurs we also construct infinite families of graphs with w r functions that are non analytic at 1 q 0 and investigate the properties of these functions our results are consistent with the conjecture that a sufficient condition for w r g q to be analytic at 1 q 0 is that g is a regular lattice graph lambda this is known not to be a necessary condition

large time behavior of schrodinger heat kernels and applications we obtain global in time and qualitatively sharp bounds for the heat kernel g of the schrodinger operator v the potential v satisfies v x c d x b near infinity with b 0 2 when v 0 our result can be described as follows g is bounded from above and below by the multiples of standard gaussians with a weight function if b 2 then the weight is bounded between two positive it constants if b 2 the weight is bounded between two positive functions of t d x and d y which have polynomial decay if b 2 or b 0 an application to a semilinear elliptic problem is also given

duality relation for potts multispin correlation functions the duality relation for multispin correlation functions of the potts model on a plane graph is derived using the formulation of a percolation average it is shown that in simple cases the correlation functions are related to those of the dual graph but more generally the correlation functions are expressible as linear combinations of ratios of certain percolation averages the dual partitioned equivalent transmissivities and dual correlation functions an overall multiplication factor arises in these considerations and the author elucidate the graph theoretic meaning of this function

chromatic polynomials for families of strip graphs and their asymptotic limits abstract we calculate the chromatic polynomials p gs m q and from these the asymptotic limiting functions w g s q lim n p g s q 1 n for families of n vertex graphs gs m comprised of m repeated subgraphs h adjoined to an initial graph i these calculations of w gs q for infinitely long strips of varying widths yield important insights into properties of w q for two dimensional lattices in turn these results connect with statistical mechanics since w q is the ground state degeneracy of the q state potts model on the lattice for our calculations we develop and use a generating function method which enables us to determine both the chromatic polynomials of finite strip graphs and the resultant w gs q function in the limit n from this we obtain the exact continuous locus of points b where w gs q is nonanalytic in the complex q plane this locus is shown to consist of arcs which do not separate the q plane into disconnected regions zeros of chromatic polynomials are computed for finite strips and compared with the exact locus of singularities b we find that as the width of the infinitely long strips is increased the arcs comprising b elongate and move toward each other which enables one to understand the origin of closed regions that result for the infinite 2d lattice

flow polynomials and their asymptotic limits for lattice strip graphs we present exact calculations of flow polynomials f g q for lattice strips of various fixed widths l y 4 and arbitrarily great lengths l x with several different boundary conditions square honeycomb and triangular lattice strips are considered we introduce the notion of flows per face fl in the infinite length limit we study the zeros of f g q in the complex q plane and determine exactly the asymptotic accumulation sets of these zeros in the infinite length limit for the various families of strips the function fl is nonanalytic on this locus the loci are found to be noncompact for many strip graphs with periodic or twisted periodic longitudinal boundary conditions and compact for strips with free longitudinal boundary conditions we also find the interesting feature that aside from the trivial case l y 1 the maximal point q cf where crosses the real axis is universal on cyclic and mobius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value q cf 3 for the infinite square lattice

semismall perturbations semi intrinsic ultracontractivity and integral representations of nonnegative solutions for parabolic equations we consider nonnegative solutions of a parabolic equation in a cylinder d i where d is a noncompact domain of a riemannian manifold and i 0 t with 0 t or i 0 under the assumption ssp i e the constant function 1 is a semismall perturbation of the associated elliptic operator on d we establish an integral representation theorem of nonnegative solutions in the case i 0 t any nonnegative solution is represented uniquely by an integral on d 0 md 0 t where md is the martin boundary of d for the elliptic operator and in the case i 0 any nonnegative solution is represented uniquely by the sum of an integral on md 0 and a constant multiple of a particular solution we also show that ssp implies the condition siu i e the associated heat kernel is semi intrinsically ultracontractive

martin boundaries of elliptic skew products semismall perturbations and fundamental solutions of parabolic equations abstract we consider positive solutions of elliptic partial differential equations on non compact domains of riemannian manifolds we explicitly determine martin boundaries and martin kernels for a class of elliptic equations in skew product form by exploiting and developing perturbation theory for elliptic equations and short long time estimates for fundamental solutions of parabolic equations

large time behavior of the heat kernel and the behavior of the green function near criticality for nonsymmetric elliptic operators abstract let p be a second order elliptic operator which is defined on a domain r n and let w x be a nonzero nonnegative function assume that p p w x admits a positive minimal green function g p x y for all lim 0 g p x y f x y where f 0 for x y unless p and p admit ground states x and x respectively with eigenvalue zero such that z w z z dz in this case f x y x y z w z z dx moreover if z z dz then the heat kernel k x y t of the parabolic equation lu u t pu 0 satisfies lim t k x y t x y z z dz

ground state entropy of potts antiferromagnets on homeomorphic families of strip graphs abstract we present exact calculations of the zero temperature partition function and ground state degeneracy per site w for the q state potts antiferromagnet on a variety of homeomorphic families of planar strip graphs g ch k1 k2 k m where k1 k2 and k describe the homeomorphic structure and m denotes the length of the strip several different ways of taking the total number of vertices to infinity by sending i m with k1 k2 and k fixed ii k1 and or k2 with m and k fixed and iii k with m and p k1 k2 fixed are studied and the respective loci of points b where w is nonanalytic in the complex q plane are determined the b s for limit i are comprised of arcs which do not enclose regions in the q plane and for many values of p and k include support for re q b for limits ii and iii is the unit circle q 1 1

ground state entropy of the potts antiferromagnet on cyclic strip graphs we present exact calculations of the zero temperature partition function chromatic polynomial and the exponent of the ground state entropy s0 for the q state potts antiferromagnet on families of cyclic and twisted cyclic mobius strip graphs composed of p sided polygons our results suggest a general rule concerning the maximal region in the complex q plane to which one can analytically continue from the physical interval where s0 0 the chromatic zeros and their accumulation set exhibit the rather unusual property of including support for re q 0 and provide further evidence for a relevant conjecture

absence of phase transition for antiferromagnetic potts models via the dobrushin uniqueness theorem we prove that theq state potts antiferromagnet on a lattice of maximum coordination numberr exhibits exponential decay of correlations uniformly at all temperatures including zero temperature wheneverq 2r we also prove slightly better bounds for several two dimensional lattices square lattice exponential decay forq 7 triangular lattice q 11 hexagonal lattice q 4 and kagome lattice q 6 the proofs are based on the dobrushin uniqueness theorem

t 0 partition functions for potts antiferromagnets on square lattice strips with twisted periodic boundary conditions we present exact calculations of the zero temperature partition function for the q state potts antiferromagnet or equivalently the chromatic polynomial for two families of arbitrarily long strip graphs of the square lattice with periodic boundary conditions in the transverse direction and i periodic and ii twisted periodic boundary conditions in the longitudinal direction so that the strip graphs are embedded on i a torus and ii a klein bottle in the limit of infinite length we calculate the exponent of the entropy w q show it to be the same for i and ii and determine its analytic structure

ground state entropy of potts antiferromagnets on cyclic polygon chain graphs we present exact calculations of chromatic polynomials for families of cyclic graphs consisting of linked polygons where the polygons may be adjacent or separated by a given number of bonds from these we calculate the exponential of the ground state entropy w for the q state potts model on these graphs in the limit of infinitely many vertices a number of properties are proved concerning the continuous locus of nonanalyticities in w our results provide further evidence for a general rule concerning the maximal region in the complex q plane to which one can analytically continue from the physical interval where s0 0

chromatic polynomials for lattice strips with cyclic boundary conditions abstract the zero temperature q state potts model partition function for a lattice strip of fixed width l y and arbitrary length l x has the form p g q j 1 n g c g j g j l x and is equivalent to the chromatic polynomial for this graph we present exact zero temperature partition functions for strips of several lattices with fbc y pbc x i e cyclic boundary conditions in particular the chromatic polynomial of a family of generalized dodecahedra graphs is calculated the coefficient c g j of degree d in q is c d u 2d q 2 where u n x is the chebyshev polynomial of the second kind we also present the chromatic polynomial for the strip of the square lattice with pbc y pbc x i e toroidal boundary conditions and width l y 4 with the property that each set of four vertical vertices forms a tetrahedron a number of interesting and novel features of the continuous accumulation set of the chromatic zeros b are found

a mapping relating complex and physical temperatures in the 2d q state potts model and some applications we show an exact equivalence of the free energy of the q state potts antiferromagnet on a lattice for the full temperature interval and the free energy of the q state potts model on the dual lattice for a semi infinite interval of complex temperatures cts this implies the existence of two quite different types of ct singularities the generic kind which does not obey universality or various scaling relations and a special kind which does obey such properties and encodes information of direct physical relevance we apply this observation to characterize ct properties of the potts model on several lattices to rule out two existing conjectures and to determine the critical value of q above which the potts antiferromagnet on the diced lattice has no phase transition

exact potts model partition functions on ladder graphs abstract we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex ladder graphs i e strips of the square lattice with width ly 2 and arbitrary length lx with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of these ladder graphs and the thermodynamics is discussed by comparison with strip graphs of other widths we analyze how the singularities at the zero temperature critical point of the ferromagnet on infinite length finite width strips depend on the width we point out and study the following noncommutativity at certain special values q s lim n lim q q s z 1 n lim q q s lim n z 1 n it is shown that the potts antiferromagnet on both the infinite length line and ladder graphs with cyclic or mobius boundary conditions exhibits a phase transition at finite temperature if 0 b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros certain properties of the complex temperature phase diagrams are shown to exhibit close connections with those of the model on the square lattice showing that exact solutions on infinite length strips provide a way of gaining insight into these complex temperature phase diagrams

complex temperature partition function zeros of the potts model on the honeycomb and kagome lattices we calculate complex temperature ct zeros of the partition function for the q state potts model on the honeycomb and kagom e lattices for several values of q these give information on the ct phase diagrams a comparison of results obtained for different boundary conditions and a discussion of some ct singularities are given among other results our findings show that the potts antiferromagnet with q 4 and q 5 on the kagom e lattice has no phase transition at either finite or zero temperature

potts model partition functions for self dual families of strip graphs abstract we consider the q state potts model on families of self dual strip graphs g d of the square lattice of width l y and arbitrarily great length l x with periodic longitudinal boundary conditions the general partition function z and the t 0 antiferromagnetic special case p chromatic polynomial have the respective forms j 1 n f l y c f l y j f l y j l x with f z p for arbitrary l y we determine i the general coefficient c f l y j in terms of chebyshev polynomials ii the number n f l y d of terms with each type of coefficient and iii the total number of terms n f l y we point out interesting connections between the n z l y d and temperley lieb algebras and between the n f l y and enumerations of directed lattice animals exact calculations of p are presented for 2 l y 4 in the limit of infinite length we calculate the ground state degeneracy per site exponent of the ground state entropy w q generalizing q from z to c we determine the continuous locus b in the complex q plane where w q is singular we find the interesting result that for all l y values considered the maximal point at which b crosses the real q axis denoted q c is the same and is equal to the value for the infinite square lattice q c 3 this is the first family of strip graphs of which we are aware that exhibits this type of universality of q c

zeros of the partition function for the triangular lattice three state potts model the authors obtain the zeros of the partition function for the isotropic triangular lattice three state potts model in a finite lattice the distribution exhibits new points on the real axis which are well fitted by an algebraic equation deduced from the inversion relation and the symmetries of the anisotropic model they identify a symmetry approximately relating all six transition points

ground state degeneracy of potts antiferromagnets homeomorphic classes with noncompact w boundaries abstract we present exact calculations of the zero temperature partition function z g q t 0 and ground state degeneracy w g q for the q state potts antiferromagnet on a number of families of graphs g for which generalizing q from z to c the boundary b of regions of analyticity of w in the complex q plane is noncompact passing through z 1 q 0 for these types of graphs since the reduced function w red q 1 w is nonanalytic at z 0 there is no large q taylor series expansion of w red the study of these graphs thus gives insight into the conditions for the validity of the large q expansions it is shown how such families of graphs can be generated from known families by homeomorphic expansion

chromatic polynomials and representations of the symmetric group abstract the chromatic polynomials considered in this paper are associated with graphs constructed in the following way take n copies of a complete graph k b and for i 1 2 n join each vertex in the ith copy to the same vertex in the i 1 th copy taking n 1 1 by convention previous calculations for b 2 and b 3 suggest that the chromatic polynomial contains terms that occur in levels in the present paper the levels are explained by using a version of the sieve principle and it is shown that the terms at level l correspond to the irreducible representations of the symmetric group sym l in the case of the two linear representations the terms can be calculated explicitly by methods based on the theory of distance regular graphs for the nonlinear representations the calculations are more complicated an illustration is given in section 10 where the complete chromatic polynomial for the case b 4 is obtained

lower bounds and series for the ground state entropy of the potts antiferromagnet on archimedean lattices and their duals we prove a general rigorous lower bound for w lambda q exp s 0 lambda q k b the exponent of the ground state entropy of the q state potts antiferromagnet on an arbitrary archimedean lattice lambda we calculate large q series expansions for the exact w r lambda q q 1 w lambda q and compare these with our lower bounds on this function on the various archimedean lattices it is shown that the lower bounds coincide with a number of terms in the large q expansions and hence serve not just as bounds but also as very good approximations to the respective exact functions w r lambda q for large q on the various lattices lambda plots of w r lambda q are given and the general dependence on lattice coordination number is noted lower bounds and series are also presented for the duals of archimedean lattices as part of the study the chromatic number is determined for all archimedean lattices and their duals finally we report calculations of chromatic zeros for several lattices these provide further support for our earlier conjecture that a sufficient condition for w r lambda q to be analytic at 1 q 0 is that lambda is a regular lattice

large time behavior of the heat kernel in this paper we study the large time behavior of the minimal heat kernel kpm x y t of a general time independent parabolic operator lu ut p x x u which is defined on a noncompact manifold m more precisely we prove that limt e 0tkpm x y t always exists here 0 is the generalized principal eigenvalue of the operator p in m

ground state entropy of the potts antiferromagnet with next nearest neighbor spin spin couplings on strips of the square lattice we present exact calculations of the zero temperature partition function chromatic polynomial and w q the exponent of the ground state entropy for the q state potts antiferromagnet with next nearest neighbor spin spin couplings on square lattice strips of width l y 3 and l y 4 vertices and arbitrarily great length l x vertices with both free and periodic boundary conditions the resultant values of w for a range of physical q values are compared with each other and with the values for the full two dimensional lattice these results give insight into the effect of such nonnearest neighbor couplings on the ground state entropy we show that the q 2 ising and q 4 potts antiferromagnets have zero temperature critical points on the l x ensuremath rightarrow ensuremath infty limits of the strips that we study with the generalization of q from z to c we determine the analytic structure of w q in the q plane for the various cases

asymptotic of the heat kernel in general benedicks domains using a new inequality relating the heat kernel and the probability of survival we prove asymptotic ratio limit theorems for the heat kernel and survival probability in general benedicks domains in particular the dimension of the cone of positive harmonic measures with dirichlet boundary conditions can be derived from the rate of convergence to zero of the heat kernel or the survival probability

exact potts model partition functions for strips of the square lattice we present exact calculations of the potts model partition function z g q v for arbitrary q and temperature like variable v on n vertex square lattice strip graphs g for a variety of transverse widths l t and for arbitrarily great length l l with free longitudinal boundary conditions and free and periodic transverse boundary conditions these have the form z g q v sum j 1 n z g lambda c z g j lambda z g j l ell we give general formulas for n z g j and its specialization to v 1 for arbitrary l t for both types of boundary conditions as well as other general structural results on z the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed it is shown how the internal energy calculated for the case of cylindrical boundary conditions is connected with critical quantities for the potts model on the infinite square lattice considering the full generalization to arbitrary complex q and v we determine the singular locus b arising as the accumulation set of partition function zeros as l l in the q plane for fixed v and in the v plane for fixed q

ground state entropy of potts antiferromagnets bounds series and monte carlo measurements we report several results concerning w ensuremath lambda q mathrm exp s 0 k b the exponent of the ground state entropy of the potts antiferromagnet on a lattice ensuremath lambda first we improve our previous rigorous lower bound on w mathrm hc q for the honeycomb hc lattice and find that it is extremely accurate it agrees to the first 11 terms with the large q series for w mathrm hc q second we investigate the heteropolygonal archimedean 4 ensuremath cdot 8 2 lattice derive a rigorous lower bound on w 4 ensuremath cdot 8 2 q and calculate the large q series for this function to o y 12 where y 1 q ensuremath 1 remarkably these agree exactly to all 13 terms calculated we also report monte carlo measurements and find that these are very close to our lower bound and series third we study the effect of non nearest neighbor couplings focusing on the square lattice with next nearest neighbor bonds

chromatic polynomials of large triangular lattices evaluating the q colourings of a lattice is equivalent to solving the q state zero temperature antiferromagnetic potts model this has recently been done exactly for an infinite triangular lattice with q real here the results are extended to the full complex q plane giving the limiting distribution of the zeros of the chromatic polynomial the results are compared with finite lattice calculations and the occurrence of isolated real zeros converging on the beraha numbers is noted

exact potts model partition function on strips of the triangular lattice abstract in this paper we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex strip graphs of width ly 2 and arbitrary length of the triangular lattice with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed considering the full generalization to arbitrary complex q and temperature we determine the singular locus b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros comparisons are made with our previous exact calculation of potts model partition functions for the corresponding strips of the square lattice our present calculations yield as special cases several quantities of graph theoretic interest

exact chromatic polynomials for toroidal chains of complete graphs abstract we present exact calculations of the partition function of the zero temperature potts antiferromagnet equivalently the chromatic polynomial for graphs of arbitrarily great length composed of repeated complete subgraphs kb with b 5 6 which have periodic or twisted periodic boundary condition in the longitudinal direction in the lx limit the continuous accumulation set of the chromatic zeros b is determined we give some results for arbitrary b including the extrema of the eigenvalues with coefficients of degree b 1 and the explicit forms of some classes of eigenvalues we prove that the maximal point where b crosses the real axis qc satisfies the inequality qc b for 2 b the minimum value of q at which b crosses the real q axis is q 0 and we make a conjecture concerning the structure of the chromatic polynomial for klein bottle strips

upper and lower bounds for the ground state entropy of antiferromagnetic potts models we derive rigorous upper and lower bounds for the ground state entropy of the q state potts antiferromagnet on the honeycomb and triangular lattices these bounds are quite restrictive especially for large q copyright ital 1997 ital the american physical society

the potts model and flows i the pair correlation function it is shown that the partition function for the a state potts model with pair interactions is related to the expected number of integer mod a flows in a percolation model the relation is generalised to the pair correlation function the resulting high temperature expansion coefficients are shown to be the flow polynomials of graph theory we also prove an observation of tsallis and levy concerning the equivalent transmissivity of a cluster

study of the potts model on the honeycomb and triangular lattices low temperature series and partition function zeros we present and analyse low temperature series and complex temperature partition function zeros for the q state potts model with q 4 on the honeycomb lattice and q 3 4 on the triangular lattice a discussion is given on how the locations of the singularities obtained from the series analysis correlate with the complex temperature phase boundary extending our earlier work we include a similar discussion for the potts model with q 3 on the honeycomb lattice and with q 3 4 on the kagom lattice

potts model and graph theory elementary exposition is given of some recent developments in studies of graphtheoretic aspects of the potts model topics discussed include graphical expansions of the potts partition function and correlation functions and their relationships with the chromatic dichromatic and flow polynomials occurring in graph theory it is also shown that the potts model realization of these classic graph theoretic problems provides alternate and direct proofs of properties established heretofore only in the context of formal graph theory

ground state degeneracy of potts antiferromagnets cases with noncompact w boundaries having multiple points at we present exact calculations of the zero temperature partition function and ground state degeneracy per site for the q state potts antiferromagnet on a number of families of graphs for which the boundary of regions of analyticity of w in the complex q plane is noncompact and has the properties that i in the plane the point z 0 is a multiple point on and ii includes support for these families are generated by the method of homeomorphic expansion our results give further insight into the conditions for the validity of large q series expansions for the reduced function

ground state entropy of potts antiferromagnets and the approach to the two dimensional thermodynamic limit we study the ground state degeneracy per site exponent of the ground state entropy w lambda l x infty times l y q for the q state potts antiferromagnet on infinitely long strips with width l y of 2d lattices lambda with free and periodic boundary conditions in the y direction denoted fbc y and pbc y we show that the approach of w to its 2d thermodynamic limit as l y increases is quite rapid for moderate values of q and l y simeq 4 w lambda l x infty times l y q is within about 5 and cal o 10 3 of the 2d value w lambda l x infty times l y infty q for fbc y and pbc y respectively the approach of w to the 2d thermodynamic limit is proved to be monotonic non monotonic for fbc y pbc y it is noted that ground state entropy determinations on infinite strips can be used to obtain the central charge for cases with critical ground states

ground state degeneracy of potts antiferromagnets on two dimensional lattices approach using infinite cyclic strip graphs the q state potts antiferromagnet on a lattice lambda exhibits nonzero ground state entropy s0 kb ln w for sufficiently large q and hence is an exception to the third law of thermodynamics an outstanding challenge has been the calculation of w sq q on the square sq lattice we present here an exact calculation of w on an infinite length cyclic strip of the square lattice which embodies the expected analytic properties of w sq q similar results are given for the kagome lattice

t 0 partition functions for potts antiferromagnets on lattice strips with fully periodic boundary conditions abstract we present exact calculations of the zero temperature partition function for the q state potts antiferromagnet equivalently the chromatic polynomial for families of arbitrarily long strip graphs of the square and triangular lattices with width ly 4 and boundary conditions that are doubly periodic or doubly periodic with reversed orientation i e of torus or klein bottle type these boundary conditions have the advantage of removing edge effects in the limit of infinite length we calculate the exponent of the entropy w q and determine the continuous locus b where it is singular we also give results for toroidal strips involving crossing subgraphs these make possible a unified treatment of torus and klein bottle boundary conditions and enable us to prove that for a given strip the locus b is the same for these boundary conditions

on bounds for dynamic programs an operator theoretical model is used to provide a general framework for establishing bounds in dynamic programming by extrapolation methods a set of bounds is found for a specified solution v of an operator equation v uv with an isotone operator u containing many known bounds as a special case by applying the general setting to standard models e g to markovian decision models new bounds are suggested for a finite horizon markov renewal decision model

transfer matrices and partition function zeros for antiferromagnetic potts models i general theory and square lattice chromatic polynomial we study the chromatic polynomials zero temperature antiferromagnetic potts model partition functions p g q for m n rectangular subsets of the square lattice with m 8 free or periodic transverse boundary conditions and n arbitrary free longitudinal boundary conditions using a transfer matrix in the fortuin kasteleyn representation in particular we extract the limiting curves of partition function zeros when n which arise from the crossing in modulus of dominant eigenvalues beraha kahane weiss theorem we also provide evidence that the beraha numbers b2 b3 b4 b5 are limiting points of partition function zeros as n whenever the strip width m is 7 periodic transverse b c or 8 free transverse b c along the way we prove that a noninteger beraha number except perhaps b10 cannot be a chromatic root of any graph

structural properties of potts model partition functions and chromatic polynomials for lattice strips abstract the q state potts model partition function equivalent to the tutte polynomial for a lattice strip of fixed width l y and arbitrary length l x has the form z g q v j 1 n z g c z g j z g j l x where v is a temperature dependent variable the special case of the zero temperature antiferromagnet v 1 is the chromatic polynomial p g q using coloring and transfer matrix methods we give general formulas for c x g j 1 n x g c x g j for x z p on cyclic and mobius strip graphs of the square and triangular lattice combining these with a general expression for the unique coefficient c z g j of degree d in q c d u 2d q 2 where u n x is the chebyshev polynomial of the second kind we determine the number of z g j s with coefficient c d in z g q v for these cyclic strips of width l y to be n z l y d 2d 1 l y d 1 1 2l y l y d for 0 d l y and zero otherwise for both cyclic and mobius strips of these lattices the total number of distinct eigenvalues z g j is calculated to be n z l y 2l y l y results are also presented for the analogous numbers n p l y d and n p l y for p g q we find that n p l y 0 n p l y 1 1 m l y 1 motzkin number n z l y 0 c l y the catalan number and give an exact expression for n p l y our results for n z l y and n p l y apply for both the cyclic and mobius strips of both the square and triangular lattices we also point out the interesting relations n z l y 2 n da tri l y and n p l y 2 n da sq l y where n da n denotes the number of directed lattice animals on the lattice we find the asymptotic growths n z l y l y 1 2 4 l y and n p l y l y 1 2 3 l y as l y some general geometric identities for potts model partition functions are also presented

families of graphs with chromatic zeros lying on circles we define an infinite set of families of graphs which we call p wheels and denote wh p n that generalize the wheel p 1 and biwheel p 2 graphs the chromatic polynomial for wh p n is calculated and remarkably simple properties of the chromatic zeros are found i the real zeros occur at q 0 1 p 1 for n p even and q 0 1 p 2 for n p odd and ii the complex zeros all lie equally spaced on the unit circle q p 1 1 in the complex q plane in the n to infty limit the zeros on this circle merge to form a boundary curve separating two regions where the limiting function w wh p q is analytic viz the exterior and interior of the above circle connections with statistical mechanics are noted