scaling limit of the loop erased random walk green s function we consider loop erased random walk lerw running between two boundary points of a square grid approximation of a planar simply connected domain the lerw green s function is the probability that the lerw passes through a given edge in the domain we prove that this probability multiplied by the inverse mesh size to the power 3 4 converges in the lattice size scaling limit to a constant times an explicit conformally covariant quantity which coincides with the hbox sle 2 green s function the proof does not use sle techniques and is based on a combinatorial identity which reduces the problem to obtaining sharp asymptotics for two quantities the loop measure of random walk loops of odd winding number about a branch point near the marked edge and a spinor observable for random walk started from one of the vertices of the marked edge

discrete riemann surfaces and the ising model we define a new theory of discrete riemann surfaces and present its basic results the key idea is to consider not only a cellular decomposition of a surface but the union with its dual discrete holomorphy is defined by a straightforward discretisation of the cauchy riemann equation a lot of classical results in riemann theory have a discrete counterpart hodge star harmonicity hodge theorem weyl s lemma cauchy integral formula existence of holomorphic forms with prescribed holonomies giving a geometrical meaning to the construction on a riemann surface we define a notion of criticality on which we prove a continuous limit theorem we investigate its connection with criticality in the ising model we set up a dirac equation on a discrete universal spin structure and we prove that the existence of a dirac spinor is equivalent to criticality

the critical z invariant ising model via dimers the periodic case we study a large class of critical two dimensional ising models namely critical z invariant ising models on periodic graphs example of which are the classical mathbb z 2 triangular and honeycomb lattice at the critical temperature fisher j math phys 7 1776 1781 1966 introduced a correspondence between the ising model and the dimer model on a decorated graph thus setting dimer techniques as a powerful tool for understanding the ising model in this paper we give a full description of the dimer model corresponding to the critical z invariant ising model we prove that the dimer characteristic polynomial is equal up to a constant to the critical laplacian characteristic polynomial and defines a harnack curve of genus 0 we prove an explicit expression for the free energy and for the gibbs measure obtained as weak limit of boltzmann measures

formulas for the coefficients of half integral weight harmonic maas forms recently bruinier and ono proved that the coefficients of certain weight 1 2 harmonic weak maas forms are given as traces of singular moduli for harmonic weak maas forms here we prove that similar results hold for the coefficients of harmonic weak maas forms of weight 3 2 k k even and weight 1 2 k k odd by extending the theta lift of bruinier funke and bruinier ono moreover we generalize these results to include twisted traces of singular moduli using earlier work of the author and ehlen on the twisted bruinier funke lift employing a general duality result between weight k and 2 k we obtain formulas for all half integral weights we also show that the non holomorphic part of the theta lift in weight 1 2 k k odd is connected to the vanishing of the special value of the l function of a certain derivative of the lifted function

the z invariant ising model via dimers the z invariant ising model baxter in philos trans r soc lond a math phys eng sci 289 1359 315 346 1978 is defined on an isoradial graph and has coupling constants depending on an elliptic parameter k when k 0 the model is critical and as k varies the whole range of temperatures is covered in this paper we study the corresponding dimer model on the fisher graph thus extending our papers boutillier and de tiliere in probab theory relat fields 147 379 413 2010 commun math phys 301 2 473 516 2011 to the fullz invariant case one of our main results is an explicit local formula for the inverse of the kasteleyn operator its most remarkable feature is that it is an elliptic generalization of boutillier and de tiliere 2011 it involves a local function and the massive discrete exponential function introduced in boutillier et al invent math 208 1 109 189 2017 this shows in particular that z invariance and not criticality is at the heart of obtaining local expressions we then compute asymptotics and deduce an explicit local expression for a natural gibbs measure we prove a local formula for the ising model free energy we also prove that this free energy is equal up to constants to that of the z invariant spanning forests of boutillier et al 2017 and deduce that the two models have the same second order phase transition in k next we prove a self duality relation for this model extending a result of baxter to all isoradial graphs in the last part we prove explicit local expressions for the dimer model on a bipartite graph corresponding to the xor version of this z invariant ising model

quadri tilings of the plane we introduce quadri tilings and show that they are in bijection with dimer models on a family of graphs r arising from rhombus tilings using two height functions we interpret a sub family of all quadri tilings called triangular quadri tilings as an interface model in dimension 2 2 assigning critical weights to edges of r we prove an explicit expression only depending on the local geometry of the graph r for the minimal free energy per fundamental domain gibbs measure this solves a conjecture of kenyon invent math 150 409 439 2002 we also show that when edges of r are asymptotically far apart the probability of their occurrence only depends on this set of edges finally we give an expression for a gibbs measure on the set of all triangular quadri tilings whose marginals are the above gibbs measures and conjecture it to be that of minimal free energy per fundamental domain

mean square values of hecke l series formed with r th order characters let l be a number field containing the r th roots of unity starting with the rankin selberg convolution of a metaplectic eisenstein series on the r fold cover of gl 2 with itself we construct a dirichlet series defined over l whose coefficients involve the r th order twists of a fixed hecke l function we then observe that a group of functional equations can be naturally associated with this construction combining this with the convexity theorem for holomorphic functions of several complex variables we show that this object as a function of two complex variables admits meromorphic continuation to 2 as an application we obtain asymptotic formulae for mean square values of the r th order twists of an arbitrary hecke l function defined over l

connection probabilities and rsw type bounds for the two dimensional fk ising model we prove russo seymour welsh type uniform bounds on crossing probabilities for the fk ising fk percolation with cluster weight q 2 model at criticality independent of the boundary conditions our proof relies mainly on smirnov s fermionic observable for the fk ising model 24 which allows us to get precise estimates on boundary connection probabilities we stay in a discrete setting in particular we do not make use of any continuum limit and our result can be used to derive directly several noteworthy properties including some new ones among which are the fact that there is no infinite cluster at criticality tightness properties for the interfaces and the existence of several critical exponents in particular the half plane one arm exponent such crossing bounds are also instrumental for important applications such as constructing the scaling limit of the ising spin field 6 and deriving polynomial bounds for the mixing time of the glauber dynamics at criticality 17

free fermion checkerboard and z invariant lattice models in statistical mechanics it is shown that the two dimensional free fermion model is equivalent to a checkerboard ising model which is a special case of the general z invariant ising model expressions are given for the partition function and local correlations in terms of those of the regular square lattice ising model corresponding results are given for the self dual potts model and the application of the methods to the three dimensional zamolodchikov model is discussed the paper ends with a discussion of the critical and disorder surfaces of the checkerboard potts model

smirnov s observable for free boundary conditions interfaces and crossing probabilities we prove convergence results for variants of smirnov s fermionic observable in the critical planar ising model in the presence of free boundary conditions one application of our analysis is a simple proof of a theorem by hongler and kytola on convergence of critical ising interfaces with plus minus free boundary conditions to dipolar sle 3 and a generalization of this result to an arbitrary number of arcs carrying plus minus or free boundary conditions another application is a computation of scaling limits of crossing probabilities in the critical fk ising model with an arbitrary number of alternating wired free boundary arcs we also deduce a new crossing formula for the spin ising model

multi spin correlation functions for the z invariant ising model abstract continuing our work 1 where a explicit formula for the two point functions of the two dimensional z invariant ising model were found i obtain here different results for the higher correlation functions and several consistency checks are done

dimers in piecewise temperleyan domains we study the large scale behavior of the height function in the dimer model on the square lattice richard kenyon has shown that the fluctuations of the height function on temperleyan discretizations of a planar domain converge in the scaling limit as the mesh size tends to zero to the gaussian free field with dirichlet boundary conditions we extend kenyon s result to a more general class of discretizations moreover we introduce a new factorization of the coupling function of the double dimer model into two discrete holomorphic functions which are similar to discrete fermions defined in smirnov proceedings of the international congress of mathematicians icm madrid spain 2006 ann math 2 172 1435 1467 2010 for temperleyan discretizations with appropriate boundary modifications the results of kenyon imply that the expectation of the double dimer height function converges to a harmonic function in the scaling limit we use the above factorization to extend this result to the class of all polygonal discretizations that are not necessarily temperleyan furthermore we show that quite surprisingly the expectation of the double dimer height function in the temperleyan case is exactly discrete harmonic for an appropriate choice of laplacian even before taking the scaling limit

nonvanishing of quadratic twists of modular l functions and applications to elliptic curves if d is square free or is a fundamental discriminant then let d d denote the kronecker character for the quadratic field q d whose fundamental discriminant is d throughout d shall denote a fundamental discriminant the d quadratic twist of f denoted f d is the newform corresponding to the twist of f by the character d in particular if gcd m d 1 then f d z n 1 d n a n q n and

conformal invariance in random cluster models i holmorphic fermions in the ising model we construct discrete holomorphic observables in the ising model at criticality and show that they have conformally covariant scaling limits as mesh of the lattice tends to zero in the sequel those observables are used to construct conformally invariant scaling limits of interfaces though ising model is often cited as a classical example of conformal invariance it seems that ours is the first paper where it is actually established

toward a conformal field theory for schramm loewner evolutions we discuss the partition function point of view for chordal schramm loewner evolutions and their relationship with correlation functions in conformal field theory both are closely related to crossing probabilities and interfaces in critical models in two dimensional statistical mechanics we gather and supplement previous results with different perspectives point out remaining difficulties and suggest directions for future studies

rank one quadratic twists of an infinite family of elliptic curves a conjecture of goldfeld implies that a positive proportion of quadratic twists of an elliptic curve e q has analytic rank 1 this assertion has been confirmed by vatsal v1 and the first author by for only two elliptic curves here we confirm this assertion for infinitely many elliptic curves e q using the heegner divisors the 3 part of the class groups of quadratic fields and a variant of the binary goldbach problem for polynomials

selmer groups of quadratic twists of elliptic curves 1 1 e y a1xy a3y x 3 a2x 2 a4x a6 where a1 a2 a3 a4 a6 z let n e denote the conductor of e j e the j invariant of e and l e s n 1 a n n s the hasse weil l function of e if e is modular then let fe z n 1 ae n q n s2 n e 1 be the associated weight 2 cusp form here 1 denotes the trivial dirichlet character throughout d will denote a square free integer and d shall denote the kronecker character for the field q d let h d denote the order of cl d the ideal class group of q d if l is prime then define h d l by

critical correlations in a z invariant inhomogeneous ising model abstract we study the critical spin spin correlation function in the z invariant inhomogeneous ising model which includes the rectangular triangular honeycomb and checkerboard lattices as special cases using quadratic difference equations we derive various explicit representations for the correlation function confirming and simplifying also a miraculous result of baxter

the critical z invariant ising model via dimers locality property we study a large class of critical two dimensional ising models namely critical z invariant ising models fisher j math phys 7 1776 1781 1966 introduced a correspondence between the ising model and the dimer model on a decorated graph thus setting dimer techniques as a powerful tool for understanding the ising model in this paper we give a full description of the dimer model corresponding to the critical z invariant ising model consisting of explicit expressions which only depend on the local geometry of the underlying isoradial graph our main result is an explicit local formula for the inverse kasteleyn matrix in the spirit of kenyon invent math 150 2 409 439 2002 as a contour integral of the discrete exponential function of mercat discrete period matrices and related topics 2002 and kenyon invent math 150 2 409 439 2002 multiplied by a local function using results of boutillier and de tiliere prob theor rel fields 147 3 4 379 413 2010 and techniques of de tiliere prob th rel fields 137 3 4 487 518 2007 and kenyon invent math 150 2 409 439 2002 this yields an explicit local formula for a natural gibbs measure and a local formula for the free energy as a corollary we recover baxter s formula for the free energy of the critical z invariant ising model baxter in exactly solved models in statistical mechanics academic press london 1982 and thus a new proof of it the latter is equal up to a constant to the logarithm of the normalized determinant of the laplacian obtained in kenyon invent math 150 2 409 439 2002

discretely holomorphic parafermions and integrable boundary conditions in two dimensional statistical models possessing a discretely holomorphic parafermion we introduce a modified discrete cauchy riemann equation on the boundary of the domain and we show that the solution of this equation yields integrable boundary boltzmann weights this approach is applied to i the square lattice o n loop model where the exact locations of the special and ordinary transitions are recovered and ii the fateev zamolodchikov spin model where a new rotation invariant integrable boundary condition is discovered for generic n

mixing times of critical two dimensional potts models we study dynamical aspects of the q state potts model on an n n box at its critical c q heat bath glauber dynamics and cluster dynamics such as swendsen wang that circumvent low temperature bottlenecks are all expected to undergo critical slowdowns in the presence of periodic boundary conditions the inverse spectral gap which in the subcritical regime is o 1 should at criticality be polynomial in n for 1 4 in accordance with the predicted discontinuous phase transition this was confirmed for q 2 the ising model by the second author and sly and for sufficiently large q by borgs et al here we show that the following holds for the critical potts model on the torus for q 3 the inverse gap of glauber dynamics is no 1 for q 4 it is at most no log n and for every q 4 in the phase coexistence regime the inverse gaps of both glauber dynamics and swendsen wang dynamics are exponential in n for free or monochromatic boundary conditions and large q we show that the dynamics at criticality is faster than on the torus unlike the ising model where free periodic boundary conditions induce similar dynamical behavior at all temperatures the inverse gap of swendsen wang dynamics is exp no 1 2017 wiley periodicals inc

the z invariant massive laplacian on isoradial graphs we introduce a one parameter family of massive laplacian operators delta m k k in 0 1 defined on isoradial graphs involving elliptic functions we prove an explicit formula for the inverse of delta m k the massive green function which has the remarkable property of only depending on the local geometry of the graph and compute its asymptotics we study the corresponding statistical mechanics model of random rooted spanning forests we prove an explicit local formula for an infinite volume boltzmann measure and for the free energy of the model we show that the model undergoes a second order phase transition at k 0 thus proving that spanning trees corresponding to the laplacian introduced by kenyon invent math 150 2 409 439 2002 are critical we prove that the massive laplacian operators delta m k k in 0 1 provide a one parameter family of z invariant rooted spanning forest models when the isoradial graph is moreover mathbb z 2 periodic we consider the spectral curve of the massive laplacian we provide an explicit parametrization of the curve and prove that it is harnack and has genus 1 we further show that every harnack curve of genus 1 with z w leftrightarrow z 1 w 1 symmetry arises from such a massive laplacian

interaural correlation sensitivity sensitivity to differences in interaural correlation was measured for 1 3 erb wide bands of noise using a 2ifc task at six frequencies 250 500 750 1000 1250 and 1500 hz the sensitivity index d was measured for discriminations between a number of fixed pairs of correlation values cumulative d functions were derived for each frequency and condition the d for discriminating any two values of correlation may be recovered from the cumulative d function by the difference between cumulative d s for these values two conditions were employed the noisebands were either presented in isolation narrow band condition or in the context of broad contiguous flanking bands of correlated noise fringed condition the cumulative d functions showed greater sensitivity to differences in correlation close to 1 than close to 0 at low frequencies but this difference was less pronounced in the fringed condition also a more linear relationship was observed when cumulative d was plotted as a function of the equivalent signal to noise ratio snr in db for each correlation value rather than directly against correlation the equivalent snr was the snr at which the interaural correlation in an nos stimulus would equal the interaural correlation of the noise used in the experiment the maximum cumulative d declined above 750 hz this decline was steeper for the fringed than for the narrow band condition for the narrow band condition the total cumulative d was variable across listeners all cumulative d functions were closely fitted using a simple two parameter function the complete data sets averaged across listeners from the fringed and narrow band conditions were fitted using functions to describe the changes in these parameters over frequency in order to produce an interpolated family of curves that describe sensitivity at frequencies between those tested these curves predict the spectra recovered by the binaural system when complex sounds such as speech are masked by noise

detection of static and dynamic changes in interaural correlation this study examines the relation between a static and a dynamic measure of interaural correlation discrimination 1 the just noticeable difference jnd in interaural correlation and 2 the minimum detectable duration of a fixed interaural correlation change embedded within a single noise burst of a given reference correlation for the first task jnds were obtained from reference interaural correlations of 1 1 and from 0 interaural correlation in either the positive or negative direction for the dynamic task duration thresholds were obtained for a brief target noise of 1 1 and 0 interaural correlation embedded in reference marker noise of 1 1 and 0 interaural correlation performance with a reference interaural correlation of 1 was significantly better than with a reference correlation of 1 similarly when the reference noise was interaurally uncorrelated discrimination was significantly better for a target correlation change towards 1 than towards 1 thus for both static and dynamic tasks interaural correlation discrimination in the positive range was significantly better than in the negative range using the two measures the length of a binaural temporal window was estimated its equivalent rectangular duration erd was approximately 86 ms and independent of the interaural correlation configuration

critical ising interfaces in multiply connected domains we prove a general result on convergence of interfaces in the critical planar ising model to conformally invariant curves absolutely continuous with respect to sle 3 our setup includes multiple interfaces on arbitrary finitely connected domains and we also treat the radial sle case in the case of simply and doubly connected domains the limiting processes are described explicitly in terms of rational and elliptic functions respectively

conformal invariance in the fk representation of the quantum ising model and convergence of the interface to the mathrm sle _ 16 3 we study the interface in the fk representation of the 1d quantum ising model and show that in the limit it converges to the mathrm sle 16 3 curve

the phase transitions of the planar random cluster and potts models with q ge 1 are sharp we prove that random cluster models with q ge 1 on a variety of planar lattices have a sharp phase transition that is that there exists some parameter p c below which the model exhibits exponential decay and above which there exists a s an infinite cluster the result may be extended to the potts model via the edwards sokal coupling our method is based on sharp threshold techniques and certain symmetries of the lattice in particular it makes no use of self duality part of the argument is not restricted to planar models and may be of some interest for the understanding of random cluster and potts models in higher dimensions due to its nature this strategy could be useful in studying other planar models satisfying the fkg lattice condition and some additional differential inequalities

local conformal structure of liouville quantum gravity liouville conformal field theory lcft is an essential building block of polyakov s formulation of non critical string theory moreover scaling limits of statistical mechanics models on planar maps are believed by physicists to be described by lcft a rigorous probabilistic formulation of lcft based on a path integral formulation was recently given by the present authors and f david in cite dkrv in the present work we prove the validity of the conformal ward identities and the belavin polyakov zamolodchikov bpz differential equations of order 2 for the correlation functions of lcft this initiates the program started in the seminal work of belavin polyakov zamolodchikov cite bpz in a probabilistic setup for a non trivial conformal field theory we also prove several celebrated results on lcft in particular an explicit formula for the 4 point correlation functions with insertion of a second order degenerate field leading to a rigorous proof of a non trivial functional relation on the 3 point structure constants derived earlier in the physics literature by teschner cite tesc the proofs are based on exact identities which rely on the underlying gaussian structure of lcft combined with estimates from the theory of critical gaussian multiplicative chaos and a careful analysis of singular integrals beurling transforms and generalizations as a by product we give bounds on the correlation functions when two points collide making rigorous certain predictions from physics on the so called operator product expansion of lcft

holomorphic spinor observables in the critical ising model we introduce a new version of discrete holomorphic observables for the critical planar ising model these observables are holomorphic spinors defined on double covers of the original multiply connected domain we compute their scaling limits and show their relation to the ratios of spin correlations thus providing a rigorous proof to a number of formulae for those ratios predicted by cft arguments

exactly solved models on planar graphs with vertices in z 3 it is shown how exactly solved edge interaction models on the square lattice may be extended onto more general planar graphs with edges connecting a subset of next nearest neighbour vertices of mathbb z 3 this is done by using local deformations of the square lattice that arise through the use of the star triangle relation similar to baxter s z invariance property these local deformations leave the partition function invariant up to some simple factors coming from the star triangle relation the deformations used here extend the usual formulation of z invariance by requiring the introduction of oriented rapidity lines which form directed closed paths in the rapidity graph of the model the quasi classical limit is also considered in which case the deformations imply a classical z invariance property as well as a related local closure relation for the action functional of a system of classical discrete laplace equations

q dependent susceptibilities in ferromagnetic quasiperiodic z invariant ising models we study the q dependent susceptibility q of a series of quasiperiodic ising models on the square lattice several different kinds of aperiodic sequences of couplings are studied including the fibonacci and silver mean sequences some identities and theorems are generalized and simpler derivations are presented we find that the q dependent susceptibilities are periodic with the commensurate peaks of q located at the same positions as for the regular ising models hence incommensurate everywhere dense peaks can only occur in cases with mixed ferromagnetic antiferromagnetic interactions or if the underlying lattice is aperiodic for mixed interaction models the positions of the peaks depend strongly on the aperiodic sequence chosen

a second quantized kolmogorov chentsov theorem via the operator product expansion we establish a direct connection between two fundamental topics one in probability theory and one in quantum field theory the first topic is the problem of pointwise multiplication of random schwartz distributions which has been the object of recent progress thanks to hairer s theory of regularity structures and the theory of paracontrolled distributions introduced by gubinelli imkeller and perkowski the second topic is wilson s operator product expansion which is a general property of models of quantum field theory and a cornerstone of the bootstrap approach to conformal field theory our main result is a general theorem for the almost sure construction of products of random distributions by mollification and suitable additive as well as multiplicative renormalizations the hypothesis for this theorem is the operator product expansion with precise bounds for pointwise correlations we conjecture these bounds to be universal features of quantum field theories with gapped dimension spectrum our theorem can accommodate logarithmic corrections anomalous scaling dimensions and even lack of translation invariance however it only applies to fields with short distance singularities that are milder than white noise as an application we provide a detailed treatment of a scalar conformal field theory of mean field type i e the fractional massless free field also known as the fractional gaussian field

heegner divisors l functions and harmonic weak maass forms recent works mostly related to ramanujan s mock theta functions make use of the fact that harmonic weak maass forms can be combinatorial generating functions generalizing works of waldspurger kohnen and zagier we prove that such forms also serve as generating functions for central values and derivatives of quadratic twists of weight 2 modular l functions to obtain these results we construct differentials of the third kind with twisted heegner divisor by suitably generalizing the borcherds lift to harmonic weak maass forms the connection with periods fourier coefficients derivatives of l functions and points in the jacobian of modular curves is obtained by analyzing the properties of these differentials using works of scholl waldschmidt and gross and zagier

conformal invariance of spin correlations in the planar ising model we rigorously prove the existence and the conformal invariance of scaling limits of the magnetization and multi point spin correlations in the critical ising model on arbitrary simply connected planar domains this solves a number of conjectures coming from the physical and the mathematical literature the proof relies on convergence results for discrete holomorphic spinor observables and probabilistic techniques

partition function of periodic isoradial dimer models isoradial dimer models were introduced in kenyon invent math 150 2 409 439 2002 they consist of dimer models whose underlying graph satisfies a simple geometric condition and whose weight function is chosen accordingly in this paper we prove a conjecture of kenyon in invent math 150 2 409 439 2002 namely that for periodic isoradial dimer models the growth rate of the toroidal partition function has a simple explicit formula involving the local geometry of the graph only this is a surprising feature of periodic isoradial dimer models which does not hold in the general periodic dimer case kenyon et al in ann math 2006

infinite conformal symmetry of critical fluctuations in two dimensions we study the massless quantum field theories describing the critical points in two dimensional statistical systems these theories are invariant with respect to the infinite dimensional group of conformal analytic transformations it is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the virasoro algebra exactly solvable theories associated with degenerate representations are analized in these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations

influence and sharp threshold theorems for monotonic measures the influence theorem for product measures on the discrete space 0 1 n may be extended to probability measures with the property of monotonicity which is equivalent to strong positive association corresponding results are valid for probability measures on the cube 0 1 n that are absolutely continuous with respect to lebesgue measure these results lead to a sharp threshold theorem for measures of random cluster type and this may be applied to box crossings in the two dimensional random cluster model

ranks of twists of elliptic curves and hilbert s tenth problem in this paper we investigate the 2 selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields we give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2 selmer rank and we give lower bounds for the number of twists with bounded conductor that have a given 2 selmer rank as a consequence under appropriate hypotheses we can find many twists with trivial mordell weil group and assuming the shafarevich tate conjecture many others with infinite cyclic mordell weil group using work of poonen and shlapentokh it follows from our results that if the shafarevich tate conjecture holds then hilbert s tenth problem has a negative answer over the ring of integers of every number field

non vanishing of quadratic twists of modular l functions k have been the subject of much study both because of their intrinsicinterest and because of the prominent role they have played inkolyvagin s work on the birch and swinnerton dyer conjecture see b f h i j ko ma m m m1 o s p p waldspurger proved a fundamental theorem the ore me 1 w1 relating these central critical values to the fourier coe cients of half integral weight cusp forms for notational convenience if d is afundamental discriminant of a quadratic number eld then de ne d

a pfaffian formula for monomer dimer partition functions we consider the monomer dimer partition function on arbitrary finite planar graphs and arbitrary monomer and dimer weights with the restriction that the only non zero monomer weights are those on the boundary we prove a pfaffian formula for the corresponding partition function as a consequence of this result multipoint boundary monomer correlation functions at close packing are shown to satisfy fermionic statistics our proof is based on the celebrated kasteleyn theorem combined with a theorem on pfaffians proved by one of the authors and a careful labeling and directing procedure of the vertices and edges of the graph

from cycle rooted spanning forests to the critical ising model an explicit construction fisher established an explicit correspondence between the 2 dimensional ising model defined on a graph g and the dimer model defined on a decorated version mathcal g of this graph fisher in j math phys 7 1776 1781 1966 in this paper we explicitly relate the dimer model associated to the critical ising model and critical cycle rooted spanning forests crsfs this relation is established through characteristic polynomials whose definition only depends on the respective fundamental domains and which encode the combinatorics of the model we first show a matrix tree type theorem establishing that the dimer characteristic polynomial counts crsfs of the decorated fundamental domain mathcal g 1 our main result consists in explicitly constructing crsfs of mathcal g 1 counted by the dimer characteristic polynomial from crsfs of g1 where edges are assigned kenyon s critical weight function kenyon in invent math 150 2 409 439 2002 thus proving a relation on the level of configurations between two well known 2 dimensional critical models

crossing probabilities for voronoi percolation we prove that the standard russo seymour welsh theory is valid for voronoi percolation this implies that at criticality the crossing probabilities for rectangles are bounded by constants depending only on their aspect ratio this result has many consequences such as the polynomial decay of the one arm event at criticality

rhombic embeddings of planar quad graphs given a finite or infinite planar graph all of whose faces have degree 4 we study embeddings in the plane in which all edges have length 1 that is in which every face is a rhombus we give a necessary and sucient condition for the existence of such an embedding as well as a description of the set of all such embeddings r etant donne un graphe planaire fini ou infini dont toutes les faces sont de degre 4 onetudie ses plongements dans le plan dont toutes les aretes sont de longueur 1 c estdire dont toutes les faces sont des losanges on donne une condition necessaire et susante pour l existence d un tel plongement et on decrit l ensemble de ces plongements

elements of class groups and shafarevich tate groups of elliptic curves the problem of estimating the number of imaginary quadratic fields whose ideal class group has an element of order l 2 is classical in number theory analogous questions for quadratic twists of elliptic curves have been the focus of recent interest whereas works of c stewart and j top st and of f gouvea and b mazur gm address the nontriviality of mordell weil groups less is known about the nontriviality of shafarevich tate groups here we obtain a new type of result for the nontriviality of class groups of imaginary quadratic fields using the circle method and then we combine it with works of g frey f v kolyvagin k and k ono o2 to obtain results on the nontriviality of shafarevich tate groups of certain elliptic curves for e x0 11 these results imply that x d 0 d fundamental and e d 5 0 x3 5log 2x

the scaling limit of the energy correlations in non integrable ising models we obtain an explicit expression for the multipoint energy correlations of a non solvable two dimensional ising models with nearest neighbor ferromagnetic interactions plus a weak finite range interaction of strength in a scaling limit in which we send the lattice spacing to zero and the temperature to the critical one our analysis is based on an exact mapping of the model into an interacting lattice fermionic theory which generalizes the one originally used by schultz mattis and lieb for the nearest neighbor ising model the interacting model is then analyzed by a multiscale method first proposed by pinson and spencer if the lattice spacing is finite then the correlations cannot be computed in closed form rather they are expressed in terms of infinite convergent power series in in the scaling limit these infinite expansions radically simplify and reduce to the limiting energy correlations of the integrable ising model up to a finite renormalization of the parameters explicit bounds on the speed of convergence to the scaling limit are derived

integrability as a consequence of discrete holomorphicity loop models it has recently been established that imposing the condition of discrete holomorphicity on a lattice parafermionic observable leads to the critical boltzmann weights in a number of lattice models remarkably the solutions of these linear equations also solve the yang baxter equations we extend this analysis for the zn model by explicitly considering the condition of discrete holomorphicity on two and three adjacent rhombi for two rhombi this leads to a quadratic equation in the boltzmann weights and for three rhombi a cubic equation the two rhombus equation implies the inversion relations the star triangle relation follows from the three rhombus equation we also show that these weights are self dual as a consequence of discrete holomorphicity this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

the size of selmer groups for the congruent number problem where np is the number of solutions of the congruence y x d2x mod p then ld s has an analytic continuation as an entire function on the complex plane the conjecture of birch and swinnerton dyer then states in particular that the rank r d of ed is equal to the order r d of ld s at s 1 this being the so called analytic rank while we cannot at present find r d in all cases we can at least determine whether or not ld 1 0 and hence conjecturally whether or not r d 0 moreover one has a functional equation for ld s which relates its values at s and 2 s via a sign change ed 1 one may deduce that 1 r d ed it would then follow from the conjecture of birch and swinnerton dyer that the rank is positive whenever ed 1 according to the calculations of birch and stephens 2 one has

planar ising magnetization field i uniqueness of the critical scaling limit the aim of this paper is to prove the following result consider the critical ising model on the rescaled grid az2 then the renormalized magnetization field a a15 8 x az2 x x seen as a random distribution i e generalized function on the plane has a unique scaling limit as the mesh size a 0 the limiting field is conformally covariant

canonical periods and congruence formulae the purpose of this article is to show how congruences between the fourier coefficients of hecke eigenforms give rise to corresponding congruences between the algebraic parts of the critical values of the associated l functions this study was initiated by b mazur in his fundamental work on the eisenstein ideal see maz77 and maz79 where it was made clear that congruences for analytic l values were closely related to the integral structure of certain hecke rings and cohomology groups the results of maz79 also showed that congruences were useful in the study of nonvanishing of l functions this idea was then further developed by stevens ste82 and rubin wiles rw82 the work of rubin and wiles in particular used congruences to study the behavior of elliptic curves in towers of cyclotomic fields a key ingredient here was a theorem of washington which states roughly that almost l values in certain families are nonzero modulo p this theme has recently been taken up again in the work of ono skinner osa osb james jam and kohnen koh97 while the earlier history was primarily concerned with cyclotomic twists the current emphasis is on families of twists by quadratic characters here one wants quantitative estimates for the number of quadratic twists of a given modular form which have nonvanishing l function at s 1 we continue this trend in the present work by using our general results to obtain a strong nonvanishing theorem for the quadratic twists of modular elliptic curves with rational points of order three this generalizes a beautiful example due to kevin james and provides new evidence for a conjecture of goldfeld gol79 it should however be pointed out that even the study of quadratic twists may be traced back to mazur the reader is urged to look at pages 212 213 of maz79 and especially at the footnote at the bottom of page 213 the theorems of davenport heilbronn dh71 and washington was78 which are crucial in this paper are both mentioned in mazur s article we want to begin by discussing the congruences that lie at the heart of this article thus let f anq n be an elliptic modular cuspform of level m and weight k 2 assume that f is a simultaneous eigenform for all the hecke operators and that a1 f 1 the l function associated to f is defined by the dirichlet series l s f ann s which converges for the real part of s sufficiently large and has analytic continuation to s c a fundamental theorem of shimura shi76 states that l s f enjoys the following algebraicity property

auditory evoked potentials to changes in interaural cross correlation the objective of this study is to analyze the representation of specific binaural stimulus features in the human brain and to compare the results to psychoacoustic data late auditory evoked potentials to changes in the interaural cross correlation iacc of continuous bandpass noise 100 2000 hz were recorded 1000 sweeps per stimulus condition were averaged in experiment 1 a brief binaural gap with a gap iacc was interposed in the noise with a reference iacc n1 and p2 amplitudes increased with increasing gap duration for uncorrelated reference iacc 0 1 0 and 0 1 0 conditions objective thresholds were comparable to psychoacoustic data for uncorrelated gap iacc 1 0 1 and 1 0 1 conditions objective thresholds were about four times larger than in psychoacoustic measurements in experiment 2 reference and test interval had the same length of 800 ms and the magnitude of change in iacc was varied objective iacc jnds are larger for an uncorrelated reference than for iaccs of 1 for the uncorre

critical ising on the square lattice mixes in polynomial time the ising model is widely regarded as the most studied model of spin systems in statistical physics the focus of this paper is its dynamic stochastic version the glauber dynamics introduced in 1963 and by now the most popular means of sampling the ising measure intensive study throughout the last three decades has yielded a rigorous understanding of the spectral gap of the dynamics on mathbb z 2 everywhere except at criticality while the critical behavior of the ising model has long been the focus for physicists mathematicians have only recently developed an understanding of its critical geometry with the advent of sle cle and new tools to study conformally invariant systems

the energy density in the planar ising model we study the critical ising model on the square lattice in bounded simply connected domains with and free boundary conditions we relate the energy density of the model to a discrete fermionic spinor and compute its scaling limit by discrete complex analysis methods as a consequence we obtain a simple exact formula for the scaling limit of the energy field one point function in terms of the hyperbolic metric this confirms the predictions originating in physics but also provides a higher precision

quenched voronoi percolation we prove that the probability of crossing a large square in quenched voronoi percolation converges to 1 2 at criticality confirming a conjecture of benjamini kalai and schramm from 1999 the main new tools are a quenched version of the box crossing property for voronoi percolation at criticality and an efron stein type bound on the variance of the probability of the crossing event in terms of the sum of the squares of the influences as a corollary of the proof we moreover obtain that the quenched crossing event at criticality is almost surely noise sensitive

universality in the 2d ising model and conformal invariance of fermionic observables it is widely believed that the celebrated 2d ising model at criticality has a universal and conformally invariant scaling limit which is used in deriving many of its properties however no mathematical proof has ever been given and even physics arguments support a priori weaker mobius invariance we introduce discrete holomorphic fermions for the 2d ising model at criticality on a large family of planar graphs we show that on bounded domains with appropriate boundary conditions those have universal and conformally invariant scaling limits thus proving the universality and conformal invariance conjectures

extended z invariance for integrable vector and face models and multi component integrable quad equations in a previous paper kels in j phys a 50 49 495202 2017 the author has established an extension of the z invariance property for integrable edge interaction models of statistical mechanics that satisfy the star triangle relation str form of the yang baxter equation ybe in the present paper an analogous extended z invariance property is shown to also hold for integrable vector models and interaction round a face irf models of statistical mechanics respectively as for the previous case of the str the z invariance property is shown through the use of local cubic type deformations of a 2 dimensional surface associated to the models which allow an extension of the models onto a subset of next nearest neighbour vertices of mathbb z 3 while leaving the partition functions invariant these deformations are permitted as a consequence of the respective ybe s satisfied by the models the quasi classical limit is also considered and it is shown that an analogous z invariance property holds for the variational formulation of classical discrete laplace equations which arise in this limit from this limit new integrable 3d consistent multi component quad equations are proposed which are constructed from a degeneration of the equations of motion for irf boltzmann weights

dimers on surface graphs and spin structures ii partition functions for dimers on closed oriented surfaces are known to be alternating sums of pfaffians of kasteleyn matrices in this paper we obtain the formula for the coefficients in terms of discrete spin structures

the laplacian and dirac operators on critical planar graphs on a periodic planar graph whose edge weights satisfy a certain simple geometric condition the discrete laplacian and dirac operators have the property that their determinants and inverses only depend on the local geometry of the graph we obtain explicit expressions for the logarithms of the normalized determinants as well as the inverses of these operators we relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron in the associated dimer and spanning tree models for which the determinants of the dirac operator and the laplacian respectively play the role of the partition function this allows us to compute the entropy and correlations in terms of the local geometry in addition we define a continuous family of special discrete holomorphic functions which via convolutions gives a general process for constructing discrete holomorphic functions and discrete harmonic functions on critical planar graphs

the self dual point of the two dimensional random cluster model is critical for q 1 we prove a long standing conjecture on random cluster models namely that the critical point for such models with parameter q 1 on the square lattice is equal to the self dual point p sd q sqrt q 1 sqrt q this gives a proof that the critical temperature of the q state potts model is equal to log 1 sqrt q for all q 2 we further prove that the transition is sharp meaning that there is exponential decay of correlations in the sub critical phase the techniques of this paper are rigorous and valid for all q 1 in contrast to earlier methods valid only for certain given q the proof extends to the triangular and the hexagonal lattices as well

the critical ising model via kac ward matrices the kac ward formula allows to compute the ising partition function on any finite graph g from the determinant of 22g matrices where g is the genus of a surface in which g embeds we show that in the case of isoradially embedded graphs with critical weights these determinants have quite remarkable properties first of all they satisfy some generalized kramers wannier duality there is an explicit equality relating the determinants associated to a graph and to its dual graph also they are proportional to the determinants of the discrete critical laplacians on the graph g exactly when the genus g is zero or one finally they share several formal properties with the ray singer overline partial torsions of the riemann surface in which g embeds

discrete complex analysis on isoradial graphs abstract we study discrete complex analysis and potential theory on a large family of planar graphs the so called isoradial ones along with discrete analogues of several classical results we prove uniform convergence of discrete harmonic measures green s functions and poisson kernels to their continuous counterparts among other applications the results can be used to establish universality of the critical ising and other lattice models

the ising magnetization exponent on z2 is 1 15 we prove that for the ising model defined on the plane mathbb z 2 at beta beta c the average magnetization under an external magnetic field h 0 behaves exactly like begin aligned langle sigma 0 rangle beta c h asymp h frac 1 15 end aligned the proof which is surprisingly simple compared to an analogous result for percolation i e that theta p p p c 5 36 o 1 on the triangular lattice kesten in commun math phys 109 1 109 156 1987 smirnov and werner in math res lett 8 5 6 729 744 2001 relies on the ghs inequality as well as the rsw theorem for fk percolation from duminil copin et al commun pure appl math 64 1165 1198 2011 the use of ghs to obtain inequalities involving critical exponents is not new in this paper we show how it can be combined with rsw to obtain matching upper and lower bounds for the average magnetization

emergent planarity in two dimensional ising models with finite range interactions the known pfaffian structure of the boundary spin correlations and more generally order disorder correlation functions is given a new explanation through simple topological considerations within the model s random current representation this perspective is then employed in the proof that the pfaffian structure of boundary correlations emerges asymptotically at criticality in ising models on mathbb z 2 with finite range interactions the analysis is enabled by new results on the stochastic geometry of the corresponding random currents the proven statement establishes an aspect of universality seen here in the emergence of fermionic structures in two dimensions beyond the solvable cases