resolution de l equation de schrodinger par integration stochastique du champ magnetique we solve the heat equation associated with a schrodinger operator with magnetic field onrd using a stochastic integral which involves only the magnetic field and does not use any potential of the field then we study a particular case on the torustd to show that our formula is no longer true and that an hypothesis like simple connexity is necessary

branched polymers and dimensional reduction we establish an exact relation between self avoiding branched polymers in d 2 continuum dimensions and the hard core continuum gas at negative activity in d dimensions we review conjectures and results on critical exponents for d 2 2 3 4 and show that they are corollaries of our result we explain the connection first proposed by parisi and sourlas between branched polymers in d 2 dimensions and the yang lee edge singularity in d dimensions

dimensional reduction for generalized continuum polymers the brydges imbrie dimensional reduction formula relates the pressure of a d dimensional gas of hard spheres to a model of d 2 dimensional branched polymers brydges and imbrie s proof was non constructive and relied on a supersymmetric localization lemma the main result of this article is a constructive proof of a more general dimensional reduction formula that contains the brydges imbrie formula as a special case central to the proof are invariance lemmas which were first introduced by kenyon and winkler for branched polymers the new dimensional reduction formulas rely on invariance lemmas for central hyperplane arrangements that are due to meszaros and postnikov several applications are presented notably dimensional reduction formulas for i non spherical bodies and ii for corrections to the pressure due to symmetry effects

exact scaling functions for self avoiding loops and branched polymers it is shown that a recently conjectured form for the critical scaling function for planar self avoiding polygons weighted by their perimeter and area also follows from an exact renormalization group flow into the branched polymer problem combined with the dimensional reduction arguments of parisi and sourlas the result is generalized to higher order multicritical points yielding exact values for all their critical exponents and exact forms for the associated scaling functions

on the upper critical dimension of lattice trees and lattice animals we give a rigorous proof of mean field critical behavior for the susceptibility 1 2 and the correlation length v 1 4 for models of lattice trees and lattice animals in two cases i for the usual model with trees or animals constructed from nearest neighbor bonds in sufficiently high dimensions and ii for a class of spread out or long range models in which trees and animals are constructed from bonds of various lengths above eight dimensions this provides further evidence that for these models the upper critical dimension is equal to eight the proof involves obtaining an infrared bound and showing that a certain square diagram is finite at the critical point and uses an expansion related to the lace expansion for the self avoiding walk

dimensional reduction for directed branched polymers dimensional reduction occurs when the critical behaviour of one system can be related to that of another system in a lower dimension we show that this occurs for directed branched polymers dbp by giving an exact relationship between dbp models in d 1 dimensions and repulsive gases at negative activity in d dimensions this implies relations between exponents of the two models d 1 d the exponent describing the singularity of the pressure and d 1 d the correlation length exponent of the repulsive gas it also leads to the relation d 1 1 d where d is the yang lee edge exponent we derive exact expressions for the number of dbp of size n in two dimensions

identity of the universal repulsive core singularity with yang lee edge criticality lattice and continuum fluid models with repulsive core interactions typically display a dominant critical type singularity on the real negative activity axis lai and fisher recently suggested mainly on numerical grounds that this repulsive core singularity is universal and in the same class as the yang lee edge singularities which arise above criticality at complex activities with positive real part a general analytic demonstration of this identification is presented here using a field theory approach with separate representations of the repulsive and attractive parts of the pair interactions

joint density for the local times of continuous time markov chains we investigate the local times of a continuous time markov chain on an arbitrary discrete state space for fixed finite range of the markov chain we derive an explicit formula for the joint density of all local times on the range at any fixed time we use standard tools from the theory of stochastic processes and finite dimensional complex calculus we apply this formula in the following directions 1 we derive large deviation upper estimates for the normalized local times beyond the exponential scale 2 we derive the upper bound in varadhan s lemma for any measurable functional of the local times and 3 we derive large deviation upper bounds for continuous time simple random walk on large subboxes of z d tending to z d as time diverges we finally discuss the relation of our density formula to the ray knight theorem for continuous time simple random walk on z which is analogous to the well known ray knight description of brownian local times

hard squares with negative activity we show that the hard square lattice gas with activity z 1 has a number of remarkable properties we conjecture that all the eigenvalues of the transfer matrix are roots of unity they fall into groups strings evenly spaced around the unit circle which have interesting number theoretic properties for example the partition function on an m n lattice with periodic boundary condition is identically 1 when m and n are coprime we provide evidence for these conjectures from analytical and numerical arguments

representation of the distributions on wiener space and stochastic calculus of variations abstract in this work we construct projections and dual projections with respect to the past of the wiener process for the cameron martin space valued distributions on the wiener space with their help we then prove on the one hand an extension of clark s formula to the representation of the scalar distributions with the extended ito integral and on the other hand we give a decomposition of the vector valued distributions by exactly calculating the kernel of the divergence operator

ising model observables and non backtracking walks this paper presents an alternative proof of the connection between the partition function of the ising model on a finite graph g and the set of non backtracking walks on g the techniques used also give formulas for spin spin correlation functions in terms of non backtracking walks the main tools used are viennot s theory of heaps of pieces and turning numbers on surfaces

mayer expansions and the hamilton jacobi equation ii fermions dimensional reduction formulas we obtain estimates on effective actions for fermionic field theories by studying the flow of a continuous renormalization group transformation for bosonic theories and statistical mechanics we establish some new formulas for mayer coefficients which are consequences of dimensional reduction

collapse transition of randomly branched polymers renormalized field theory we present a minimal dynamical model for randomly branched isotropic polymers and we study this model in the framework of renormalized field theory for the swollen phase we show that our model provides a route to understand the well established dimensional reduction results from a different angle for the collapse transition we uncover a hidden becchi rouet stora supersymmetry signaling the sole relevance of tree configurations we correct the long standing one loop results for the critical exponents and we push these results on to two loop order for the collapse transition we find a runaway of the renormalization group flow which lends credence to the possibility that this transition is a fluctuation induced first order transition our dynamical model allows us to calculate for the first time the fractal dimension of the shortest path on randomly branched polymers in the swollen phase as well as at the collapse transition and related fractal dimensions

n matrix completion problem abstract an n n matrix is called an n matrix if all principal minors are negative in this paper we are interested in n matrix completion problems that is when a partial n matrix has an n matrix completion in general a combinatorially or non combinatorially symmetric partial n matrix does not have an n matrix completion here we prove that a combinatorially symmetric partial n matrix has an n matrix completion if the graph of its specified entries is a 1 chordal graph we also prove that there exists an n matrix completion for a partial n matrix whose associated graph is an undirected cycle

directed lattice animals and the lee yang edge singularity the exponents nu perpendicular to gamma of directed animals in d dimensions are shown to equal those for the lee yang problem in d 1 dimensions for d 2 the author obtains the exact results nu perpendicular to theta 1 2 and for d or 7 the scaling relation theta d 1 nu perpendicular to

dimensional reduction formulas for branched polymer correlation functions in bi01 we have proven that the generating function for self avoiding branched polymers in d 2 continuum dimensions is proportional to the pressure of the hard core continuum gas at negative activity in d dimensions this result explains why the critical behavior of branched polymers should be the same as that of the i 3 or yang lee edge field theory in two fewer dimensions as proposed by parisi and sourlas in 1981 in this article we review and generalize the results of bi01 we show that the generating functions for several branched polymers are proportional to correlation functions of the hard core gas we derive ward identities for certain branched polymer correlations we give reduction formulae for multi species branched polymers and the corresponding repulsive gases finally we derive the massive scaling limit for the 2 point function of the one dimensional hard core gas and thereby obtain the scaling form of the 2 point function for branched polymers in three dimensions

computer simulations of randomly branching polymers annealed versus quenched branching structures we present computer simulations of three systems of randomly branching polymers in d 3 dimensions ideal trees and self avoiding trees with annealed and quenched connectivities in all cases we performed a detailed analysis of trees connectivities spatial conformations and statistical properties of linear paths on trees and compare the results to the corresponding predictions of flory theory we confirm that overall the theory predicts correctly that trees with quenched ideal connectivity exhibit less overall swelling in good solvent than corresponding trees with annealed connectivity even though they are more strongly stretched on the path level at the same time we emphasize the inadequacy of the flory theory in predicting the behaviour of other and equally relevant observables like contact probabilities between tree nodes we show then that contact probabilities can be aptly characterized by introducing a novel critical exponent which accounts for how they decay as a function of the node to node path distance on the tree

on characterizing n matrices using linear complementarity we show that a square matrix a with at least one positive entry and all principal minors negative can be characterized in terms of the number of solutions the linear complementarity problem q a with the matrix a has for different vectors q such a matrix can also be characterized in terms of a sign nonreversal property these results complement the known results for a square matrix a 0 with all its principal minors negative

rooted spiral trees on hyper cubic lattices we study rooted spiral trees in 2 3 and 4 dimensions on a hyper cubic lattice using exact enumeration and monte carlo techniques on the square lattice we also obtain exact lower bound of 1 93565 on the growth constant series expansions give 1 3667 0 0010 and 0 6574 0 0010 on a square lattice with monte carlo simulations we get the estimates as 1 364 0 010 and 0 656 0 010 these results are numerical evidence against earlier proposed dimensional reduction by four in this problem in dimensions higher than two the spiral constraint can be implemented in two ways in either case our series expansion results do not support the proposed dimensional reduction

spanning forests and the q state potts model in the limit q 0 we study the q state potts model with nearest neighbor coupling v e j 1 in the limit q v 0 with the ratio w v q held fixed combinatorially this limit gives rise to the generating polynomial of spanning forests physically it provides information about the potts model phase diagram in the neighborhood of q v 0 0 we have studied this model on the square and triangular lattices using a transfer matrix approach at both real and complex values of w for both lattices we have computed the symbolic transfer matrices for cylindrical strips of widths 2 l 10 as well as the limiting curves b of partition function zeros in the complex w plane for real w we find two distinct phases separated by a transition point w w0 where w0 1 4 resp w0 0 1753 0 0002 for the square resp triangular lattice for w w0 we find a non critical disordered phase that is compatible with the predicted asymptotic freedom as w for w w0 our results are compatible with a massless berker kadanoff phase with central charge c 2 and leading thermal scaling dimension xt 1 2 marginally irrelevant operator at w w0 we find a first order critical point the first derivative of the free energy is discontinuous at w0 while the correlation length diverges as w w0 and is infinite at w w0 the critical ehavior at w w0 seems to be the same for both lattices and it differs from that of the berker kadanoff phase our results suggest that the central charge is c 1 the leading thermal scaling dimension is xt 1 0 and the critical exponents are 1 d 1 2 and 1

the scaling limit of loop erased random walk in three dimensions we show that the scaling limit exists and is invariant under dilations and rotations we give some tools that might be useful to show universality

on the universality of the nonphase transition singularity in hard particle systems activity series for hard particle lattice gases and hard particles in continuous space are examined with respect to the singularity on the negative activity axis for approximately spherical particles it is found that the nature of the singularity depends only on the dimensionality of space

cluster and virial expansions for the multi species tonks gas we consider a mixture of non overlapping rods of different lengths ell k moving in mathbb r or mathbb z our main result are necessary and sufficient convergence criteria for the expansion of the pressure in terms of the activities z k and the densities rho k this provides an explicit example against which to test known cluster expansion criteria and illustrates that for non negative interactions the virial expansion can converge in a domain much larger than the activity expansion in addition we give explicit formulas that generalize the well known relation between non overlapping rods and labelled rooted trees we also prove that for certain choices of the activities the system can undergo a condensation transition akin to that of the zero range process the key tool is a fixed point equation for the pressure

the number and size of branched polymers in high dimensions we consider two models of branched polymers lattice trees on thed dimensional hypercubic lattice i the nearest neighbor model in sufficiently high dimensions and ii a spread out or long range model ford 8 in which trees are constructed from bonds of length less than or equal to a large parameterl we prove that for either model the critical exponent for the number of branched polymers exists and equals 5 2 and that the critical exponentv for the radius of gyration exists and equals 1 4 this improves our earlier results for the corresponding generating functions the proof uses the lace expansion together with an analysis involving fractional derivatives which has been applied previously to the self avoiding walk in a similar context

generalized approach to the non backtracking lace expansion the lace expansion is a powerful perturbative technique to analyze the critical behavior of random spatial processes such as the self avoiding walk percolation and lattice trees and animals the non backtracking lace expansion noble is a modification that allows us to improve its applicability in the nearest neighbor setting on the mathbb z d lattice for percolation lattice trees and lattice animals the noble gives rise to a recursive formula that we study in this paper at a general level we state assumptions that guarantee that the solution of this recursive formula satisfies the infrared bound in two related papers we show that these conditions are satisfied for percolation in d ge 11 for lattice trees in d ge 16 and for lattice animals in d ge 18

the asymptotic evaluation of a class of path integrals ii the asymptotic behavior of a class of wiener like path integrals functions of the local time is determined these integrals are of interest in themselves and also arise very naturally in the theory of disordered systems we show that by making use of a grassman algebra i e a set of anticommuting variables the earlier treatment of this problem can be greatly simplified in particular the previous use of replica trick which involves a difficult to justify analytic continuation in the number of field components is thus avoided

critical two point functions and the lace expansion for spread out high dimensional percolation and related models we consider spread out models of self avoiding walk bond percolation lattice trees and bond lattice animals on mathbb z d having long finite range connections above their upper critical dimensions d 4 self avoiding walk d 6 percolation and d 8 trees and animals the two point functions for these models are respectively the generating function for self avoiding walks from the origin to x in mathbb z d the probability of a connection from 0 to x and the generating function for lattice trees or lattice animals containing 0 and x we use the lace expansion to prove that for sufficiently spread out models above the upper critical dimension the two point function of each model decays at the critical point as a multiple of x 2 d as x to infty we use a new unified method to prove convergence of the lace expansion the method is based on x space methods rather than the fourier transform our results also yield unified and simplified proofs of the bubble condition for self avoiding walk the triangle condition for percolation and the square condition for lattice trees and lattice animals for sufficiently spread out models above the upper critical dimension