Search

For general antiferromagnetic 2-spin systems, including the hardcore model and the antiferromagnetic Ising model, there is an $\mathsf{FPTAS}$ for the partition function on graphs of maximum degree $\Delta$ when the infinite regular tree lies in the uniqueness region by Li et al. (2013). Moreover, in the tree non-uniqueness region, Sly (2010) showed that there is no $\mathsf{FPRAS}$ to estimate the partition function unless $\mathsf{NP}=\mathsf{RP}$. The algorithmic results follow from the correlation decay approach due to Weitz (2006) or the polynomial interpolation approach developed by Barvinok (2016). However the running time is only polynomial for constant $\Delta$. For the hardcore model, recent work of Anari et al. (2020) establishes rapid mixing of the simple single-site Markov chain known as the Glauber dynamics in the tree uniqueness region. Our work simplifies their analysis of the Glauber dynamics by considering the total pairwise influence of a fixed vertex $v$ on other vertices, as opposed to the total influence on $v$, thereby extending their work to all 2-spin models and improving the mixing time. More importantly our proof ties together the three disparate algorithmic approaches: we show that contraction of the tree recursions with a suitable potential function, which is the primary technique for establishing efficiency of Weitz's correlation decay approach and Barvinok's polynomial interpolation approach, also establishes rapid mixing of the Glauber dynamics. We emphasize that this connection holds for all 2-spin models (both antiferromagnetic and ferromagnetic), and existing proofs for correlation decay or polynomial interpolation immediately imply rapid mixing of Glauber dynamics. Our proof utilizes that the graph partition function divides that of Weitz's self-avoiding walk trees, leading to new tools for analyzing influence of vertices., Comment: Section 7 and Appendix E are updated to fix a small error

A natural problem in the context of the coupon collector's problem is the behavior of the maximum of independent geometrically distributed random variables (with distinct parameters). This question has been addressed by Brennan et al. (British J. of Math. & CS. 8 (2015), 330-336). Here we provide explicit asymptotic expressions for the moments of that maximum, as well as of the maximum of exponential random variables with corresponding parameters. We also deal with the probability of each of the variables being the maximal one. The calculations lead to expressions involving Hurwitz's zeta function at certain special points. We find here explicitly the values of the function at these points. Also, the distribution function of the maximum we deal with is closely related to the generating function of the partition function. Thus, our results (and proofs) rely on classical results pertaining to the partition function., 32 pages, 2 tables

We revisit the Riemann-Hilbert problem determined by Donaldson-Thomas invariants for the resolved conifold and for other small crepant resolutions. While this problem can be recast as a system of TBA-type equations in the conformal limit, solutions are ill-defined due to divergences in the sum over infinite trajectories in the spectrum of D2-D0-brane bound states. We explore various prescriptions to make the sum well-defined, show that one of them reproduces the existing solution in the literature, and identify an alternative solution which is better behaved in a certain limit. Furthermore, we show that a suitable asymptotic expansion of the $\tau$ function reproduces the genus expansion of the topological string partition function for any small crepant resolution. As a by-product, we conjecture new integral representations for the triple sine function, similar to Woronowicz' integral representation for Faddeev's quantum dilogarithm., Comment: 25+14 pages; added proof for integral representation of the double sine function and a similar conjecture for the triple sine; version accepted for publication in Annales Henri Poincar\'e

Counting the independent sets of a graph is a classical #P-complete problem, even in the bipartite case. We give an exponential-time approximation scheme for this problem which is faster than the best known algorithm for the exact problem. The running time of our algorithm on general graphs with error tolerance $\varepsilon$ is at most $O(2^{0.2680n})$ times a polynomial in $1/\varepsilon$. On bipartite graphs, the exponential term in the running time is improved to $O(2^{0.2372n})$. Our methods combine techniques from exact exponential algorithms with techniques from approximate counting. Along the way we generalise (to the multivariate case) the FPTAS of Sinclair, Srivastava, \v{S}tefankovi\v{c} and Yin for approximating the hard-core partition function on graphs with bounded connective constant. Also, we obtain an FPTAS for counting independent sets on graphs with no vertices with degree at least 6 whose neighbours' degrees sum to 27 or more. By a result of Sly, there is no FPTAS that applies to all graphs with maximum degree 6 unless $\mbox{P}=\mbox{NP}$., Comment: 52pp

George Andrews and Ae Ja Yee recently established beautiful results involving bivariate generalizations of the third-order mock theta functions $$\omega (q)$$ and $$\nu (q)$$ , thereby extending their earlier results with the second author. Generalizing the Andrews–Yee identities for trivariate generalizations of these mock theta functions remained a mystery, as pointed out by Li and Yang in their recent work. We partially solve this problem and generalize these identities. Several new as well as well-known results are derived. For example, one of our two main theorems gives, as a corollary, a special case of Soon-Yi Kang’s three-variable reciprocity theorem. A relation between a new restricted overpartition function $$p^{*}(n)$$ and a weighted partition function $$p_*(n)$$ is obtained from one of the special cases of our second theorem.

We look for the origins of the single equation, which is a peculiar combination of W-constrains, which provides the non-abelian W-representation for generalized Kontsevich model (GKM), i.e. is enough to fix the partition function unambiguously. Namely we compare it with the scalar projection of the matrix Ward identity. It turns out that, though similar, the two equations do not coincide, moreover, the latter one is non-polynomial in time-variables. This discrepancy disappears for the cubic model if partition function is reduced to depend on odd times (belong to KdV sub-hierarchy of KP), but in general such reduction is not enough. We consider the failure of such direct interpretation of the "single equation" as a new kind of anomaly, which should be explained and eliminated in the future analysis of GKM., to Andrei Mironov on occasion of his 60's birthday; 12 pages

In this work, the atomic parameters of Ag XLIV (Be-like Ag) are examined and evaluated by implementing the GRASP2K package with the multi-configuration Dirac–Hartree–Fock (MCDHF) method for the calculation of wave-functions. We have listed fine structure energy levels of the lowest 170 levels with radiative data for multipole moments, such as electric dipole (E1), electric quadrupole (E2), magnetic dipole (M1), and magnetic quadrupole (M2) transitions, that lie in the region of extreme ultraviolet (EUV) and soft X-ray (SXR) for Ag XLIV from the ground state within the lowest 170 levels. We have compared our GRASP2K and FAC results with theoretical results available in the literature for some levels. Additionally, we have also calculated partition function and thermodynamic quantities for temperature ranges from 104 to 107 K. We believe that our presented details and data may be beneficial not only in plasma modeling but also in imaging of nanostructure as well as in medicine, semiconductors, and EUV and SXR laser applications.

Topological phases in non-Hermitian systems have become fascinating subjects recently. In this paper, we attempt to classify topological phases in 1D interacting non-Hermitian systems. We begin with the non-Hermitian generalization of the Su-Schrieffer-Heeger (SSH) model and discuss its many-body topological Berry phase, which is well defined for all interacting quasi-Hermitian systems (non-Hermitian systems that have real energy spectrum). We then demonstrate that the classification of topological phases for quasi-Hermitian systems is exactly the same as their Hermitian counterparts. Finally, we construct the fixed point partition function for generic 1D interacting non-Hermitian local systems and find that the fixed point partition function still has a one-to-one correspondence to their Hermitian counterparts. Thus, we conclude that the classification of topological phases for generic 1D interacting non-Hermitian systems is still exactly the same as Hermitian systems., 15 pages, 5 figures

The partition crank is a statistic on partitions introduced by Freeman Dyson to explain Ramanujan's congruences. In this paper, we prove that the crank is asymptotically equidistributed modulo Q, for any odd number Q. To prove this, we obtain effective bounds on the error term from Zapata Rolon's asymptotic estimate for the crank function. We then use those bounds to prove the surjectivity and strict log-subadditivity of the crank function., 14 pages 1) Some clarifications and corrections have been made. 2) This paper will appear in Ramanujan Journal

Using the relationship between Siegel cusp forms of degree $2$ and cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}})$, we derive some congruences involving dimensions of spaces of Siegel cusp forms of degree $2$ and the class number of $\mathbb{Q}(\sqrt{-p})$. We also obtain some congruences between the $4$-core partition function $c_4(n)$ and dimensions of spaces of Siegel cusp forms of degree $2$., Comment: Some minor changes. Published online in The Ramanujan Journal

We present a comprehensive and elaborate study of W LVI (K-like W55+) by using multi-configuration Dirac–Fock method (MCDF). We have included relativistic corrections, QED (quantum electrodynamics) and Breit corrections in our computation. We have reported energy levels and radiative data for multipole transitions (i.e., electric dipole (E1), electric quadrupole (E2), magnetic dipole (M1), and magnetic quadrupole (M2)) within the lowest 142 fine-structure levels and predicted soft X-ray transition (SXR) and extreme ultraviolet transitions (EUV) from higher excited states to the ground state. We have compared our calculated data with energy levels compiled by NIST and other available results in the literature and the small discrepancies found with them are discussed. Because only a few of the lowest levels are available in the literature, for checking excitation energies of higher excited states we have performed the same calculations with the distorted wave method. Furthermore, we have also provided relative population for the first five excited states, with both the partition function and thermodynamic quantities for W LVI and studied their variations with temperature. We believe that our reported new atomic data of W LVI may be useful in identification and analysis of spectral lines from various astrophysical and fusion plasma sources and may also be beneficial in plasma modeling.

The sliding electrical contact interface will generate a huge total heat source including Joule heat and frictional heat when an armature slides along a pair of rails driven by Lorentz forces. The total heat is partitioned at the contact interface flowing into armature and rail respectively. The heat partition at the rail-armature interface and the resulting temperature rise have important effects on the melt wear characteristics of the contact material. A computational method was proposed for determining heat partition at the rail-armature interface based on least squares regression. Temperatures equations of rail and armature were developed in terms of an unknown heat partition function. Solution equations of heat partition function were derived by matching the average temperature of rail and armature at the contact surfaces. The heat partition function was assumed to be a polynomial form and an estimate for the heat partition function was obtained by minimizing the error term in the least squares sense. Then, effects of polynomial order on the heat partition function are analyzed. Furthermore, the heat partition solution from the least squares regression and previous analytical method are compared. Variation characteristics of heat partition coefficient were also analyzed. In the end, the variation mechanism and influencing factors of heat partition coefficient were discussed.

We generalize the asymptotic estimates by Bubboloni, Luca and Spiga (2012) on the number of $k$-compositions of $n$ satisfying some coprimality conditions. We substantially refine the error term concerning the number of $k$-compositions of $n$ with pairwise relatively prime summands. We use a different approach, based on properties of multiplicative arithmetic functions of $k$ variables and on an asymptotic formula for the restricted partition function., Comment: revised

We study resurgence for some 3-manifold invariants when $G_{\mathbb{C}}=SL(2, \mathbb{C})$. We discuss the case of an infinite family of Seifert manifolds for general roots of unity and the case of the torus knot complement in $S^3$. Via resurgent analysis, we see that the contribution from the abelian flat connections to the analytically continued Chern-Simons partition function contains the information of all non-abelian flat connections, so it can be regarded as a full partition function of the analytically continued Chern-Simons theory on 3-manifolds $M_3$. In particular, this directly indicates that the homological block for the torus knot complement in $S^3$ is an analytic continuation of the full $G=SU(2)$ partition function, i.e. the colored Jones polynomial., Comment: 42 pages, 4 figures

We study the divisibility properties of the partition function associated with the eighth-order mock theta function $$V_0(q)$$ , introduced by Gordon and McIntosh. We obtain congruences modulo powers of 2 for certain coefficients of the partition function, akin to Ramanujan’s partition congruences. Further, we also present several infinite families of congruences modulo 13, 25, and 27.

For any positive integers $n$ and $r$, let $p_r(n)$ denotes the number of partitions of $n$ where each part has $r$ distinct colours. Many authors studied the partition function $p_r(n)$ for particular values of $r$. In this paper, we prove some general congruences modulo $5$ and $7$ for the colour partition function $p_r(n)$ by considering some general values of $r$. To prove the congruences we employ some $q$-series identities which is also in the spirit of Ramanujan., 10 Pages

The 3D-RISM-KH molecular solvation theory is applied to liquid long chain unsaturated oleyl alcohol to explore the structure of the liquid and to calculate alcohol/water partition coefficients. The united atom TraPPE force field parameters resulted in comparable structural information between RISM and MD simulations. The RISM theory and the MD simulation point to a relatively ordered liquid structure of oleyl alcohol. The molecular partition coefficients were computed with a root mean square deviation of 0.48 unit via a predictive quantitative structure activity relationship.

The purpose of this paper is to study which coalition structures have stable distributions. We employ the projective core as a stability concept. Although the projective core is often defined only for the grand coalition, we define it for every coalition structure. We apply the core notion to a variety of economic models including the public goods game, the Cournot and Bertrand competition, and the common pool resource game. We use a partition function to formulate these models. We argue that symmetry is a common property of these models in terms of a partition function. We offer some general results that hold for all symmetric partition function form games and discuss their implications in the economic models. We also provide necessary and sufficient conditions for the projective core of the models to be nonempty. In addition, we show that our results hold even in the presence of small perturbations of symmetry.

We propose a field-theoretic interpretation of Ruelle zeta function and show how it can be seen as the partition function forBFtheory when an unusual gauge-fixing condition on contact manifolds is imposed. This suggests an alternative rephrasing of a conjecture due to Fried on the equivalence between Ruelle zeta function and analytic torsion, in terms of homotopies of Lagrangian submanifolds., Annales Henri Poincaré, 21 (12), ISSN:1424-0661, ISSN:1424-0637

Recently, Andrews defined a partition function $\mathcal{EO}(n)$ which counts the number of partitions of $n$ in which every even part is less than each odd part. He also defined a partition function $\overline{\mathcal{EO}}(n)$ which counts the number of partitions of $n$ enumerated by $\mathcal{EO}(n)$ in which only the largest even part appears an odd number of times. Andrews proposed to undertake a more extensive investigation of the properties of $\overline{\mathcal{EO}}(n)$. In this article, we prove infinite families of congruences for $\overline{\mathcal{EO}}(n)$. We next study parity properties of $\overline{\mathcal{EO}}(n)$. We prove that there are infinitely many integers $N$ in every arithmetic progression for which $\overline{\mathcal{EO}}(N)$ is even; and that there are infinitely many integers $M$ in every arithmetic progression for which $\overline{\mathcal{EO}}(M)$ is odd so long as there is at least one. Very recently, Uncu has treated a different subset of the partitions enumerated by $\mathcal{EO}(n)$. We prove that Uncu's partition function is divisible by $2^k$ for almost all $k$. We use arithmetic properties of modular forms and Hecke eigenforms to prove our results., Accepted in Journal of Number Theory

In this paper, we investigated the influence of energy-dependent potentials on the thermodynamic properties of the Klein-Gordon oscillator(KGO): in this way all thermal properties have been determinate via the well-know Euler-Maclaurin method. After this, we extend our study to the case of superstatistical properties of our problem in question. The probability densityf(β)followsχ2− superstatistics (=Tsallis statistics or Gamma distribution). Under the approximation of the low-energy asymptotics of superstatistics, the partition function, at first, has been calculated. This approximation leads to a universal parameterqfor any superstatistics, not only for Tsallis statistics. By using the desired partition function, all thermal properties have been obtained in terms of the parameterq. Also, the influence of the this type of potentials on these properties, via the parameterγ, are well discussed.

In this work we build the foundations of a quantum Monte Carlo as a stochastic numerical method to solve lattice many-body quantum systems with nearest-neighbor interactions at most. As motivation, we briefly describe the bilinear-biquadratic Heisenberg model with an external field, for spin-1 particles, as an effective Hamiltonian of the Bose-Hubbard model with an external quadratic Zeeman field in the Mott insulator phase at unit filling. Then, we discuss how to implement the world line Monte Carlo with local updates to circumvent the difficulties that arise on these type of systems by mapping the quantum partition function into the one of an effective classical model, in one additional dimension, given by the imaginary time evolution of the system. Such a mapping is performed by means of the Suzuki-Trotter decomposition, which transforms the original partition function into a summation of weights given by the classical configurations. Later, we present a set of observables that can be measured through this method and show how to use a Metropolis update scheme to accomplish the measurements. At last, we present the maximization of the configuration weights for three parameter sets as the first and relevant step to perform future measurements.

Let $$(m_1,\ldots ,m_J)$$ and $$(r_1,\ldots ,r_J)$$ be two sequences of J positive integers satisfying $$1\le r_j< m_j$$ for all $$j=1,\ldots ,J$$ . Let $$(\delta _1,\ldots ,\delta _J)$$ be a sequence of J nonzero integers. In this paper, we study the asymptotic behavior of the Taylor coefficients of the infinite product $$\begin{aligned} \prod _{j=1}^J\Bigg (\prod _{k\ge 1}\big (1-q^{r_j+m_j(k-1)}\big )\big (1-q^{-r_j+m_jk}\big )\Bigg )^{\delta _j}. \end{aligned}$$ Our work generalizes many known results, including an asymptotic formula due to Lehner for the partition function arising from the first Rogers–Ramanujan identity. The main technique used here is based on Rademacher’s circle method.

We consider the limit behavior of partition function of directed polymers in random environment, which is represented by a linear model instead of a family of i.i.d.variables in 1 + 1 dimensions. Under the assumption on the environment that its spatial correlation decays algebraically, using the method developed in Alberts et al. (2014), we show that the scaled partition function, as a process defined on [ 0 , 1 ] × R , converges weakly to the solution to some stochastic heat equations driven by fractional Brownian field. The fractional Hurst parameter is determined by the correlation exponent of the random environment. Here multiple Ito integral with respect to fractional Gaussian field and spectral representation of stationary process are heavily involved.

We consider a univariate beta integral composed from general modular quantum dilogarithm functions and prove its exact evaluation formula. It represents the partition function of a particular $3d$ supersymmetric field theory on the general squashed lens space. Its possible applications to $2d$ conformal field theory are briefly discussed as well., Comment: typos corrected, 20 pages

Supersymmetry allows a $D^4 R^4$ interaction in M-theory, but such an interaction is inconsistent with string theory dualities and so is known to be absent. We provide a novel proof of the absence of the $D^4 R^4$ M-theory interaction by calculating 4-point scattering amplitudes of 11d supergravitons from ABJM theory. This calculation extends a previous calculation performed to the order corresponding to the $R^4$ interaction. The new ingredient in this extension is the interpretation of the fourth derivative of the mass deformed $S^3$ partition function of ABJM theory, which can be determined using supersymmetric localization, as a constraint on the Mellin amplitude associated with the stress tensor multiplet 4-point function. As part of this computation, we relate the 4-point function of the superconformal primary of the stress tensor multiplet of any 3d ${\cal N} = 8$ SCFT to some of the 4-point functions of its superconformal descendants. We also provide a concise formula for a general integrated 4-point function on $S^d$ for any $d$., Comment: 54 pages, Mathematica notebook attached; v2 typos fixed

Promoter is an important functional elements of DNA sequences, which is in charge of gene transcription initiation. Recognizing promoter have important help for understanding the relative life phenomena. Based on the concept that promoter is mainly determined by its sequence and structure, a novel statistical physics model for predicting promoter in Escherichia coli K-12 is proposed. The total energies of DNA local structure of sequence segments in the three benchmark promoter sequence datasets, the sole prediction parameter, are calculated by using principles from statistical physics and information theory. The better results are obtained. And a web-server PhysMPrePro for predicting promoter is established at http://202.207.14.87:8032/bioinformation/PhysMPrePro/index.asp, so that other scientists can easily get their desired results by our web-server.

The path integral for the partition function of Chern–Simons gauge theory with a compact gauge group is evaluated on a general Seifert $3$-manifold. This extends previous results and relies on abelianisation, a background field method and local application of the Kawasaki Index theorem.

We show that the fermionic matrix model can be realized by $W$-representation. We construct the Virasoro constraints with higher algebraic structures, where the constraint operators obey the Witt algebra and null 3-algebra. The remarkable feature is that the character expansion of the partition function can be easily derived from such Virasoro constraints. It is a $\tau$-function of the KP hierarchy. We construct the fermionic Aristotelian tensor model and give its $W$-representation. Moreover, we analyze the fermionic red tensor model and present the $W$-representation and character expansion of the partition function., Comment: 20 pages. Revised version accepted for publication in Nuclear Physics B

In Alexandrov's work [4,5] it has been shown that the extended partition function exp⁡(Fo,ext+Fc) introduced by Buryak in [14,15] is a tau-function of the KP hierarchy. In this work, we compute the affine coordinates of this tau-function on the Sato Grassmannian, and rewrite the Virasoro constraints as recursions for the affine coordinates in the fermionic picture. As applications we derive some formulas for the extended partition function and the connected n-point functions using methods developed by Zhou in [48] based on the boson-fermion correspondence.

For an arbitrary solution to the Burgers--KdV hierarchy, we define the tau-tuple $(\tau_1,\tau_2)$ of the solution. We show that the product $\tau_1\tau_2$ admits Buryak's residue formula. Therefore, according to Alexandrov's theorem, $\tau_1\tau_2$ is a tau-function of the KP hierarchy. We then derive a formula for the affine coordinates for the point of the Sato Grassmannian corresponding to the tau-function $\tau_1\tau_2$ explicitly in terms of those for $\tau_1$. Applications to the analogous open extension of the generalized BGW tau-function and to the open partition function are given., Comment: Minor changes: improved presentations; corrected typos; added references. 21pages

We study the Euclidean path integral of two-dimensional quantum gravity with positive cosmological constant coupled to conformal matter with large and positive central charge. The problem is considered in a semiclassical expansion about a round two-sphere saddle. We work in the Weyl gauge whereby the computation reduces to that for a (timelike) Liouville theory. We present results up to two-loops, including a discussion of contributions stemming from the gauge fixing procedure. We exhibit cancelations of ultraviolet divergences and provide a path integral computation of the central charge for timelike Liouville theory. Combining our analysis with insights from the DOZZ formula we are led to a proposal for an all orders result for the two-dimensional gravitational partition function on the two-sphere., 20 pages + appendices

We discuss two-dimensional quantum gravity coupled to conformal matter and fixed area in a semiclassical large and negative matter central charge limit. In this setup the gravity theory -- otherwise highly fluctuating -- admits a round two-sphere saddle. We discuss the two-sphere partition function up to two-loop order from the path integral perspective. This amounts to studying Feynman diagrams incorporating the fixed area constraint on the round two-sphere. In particular we find that all ultraviolet divergences cancel to this order. We compare our results with the two-sphere partition function obtained from the DOZZ formula., Comment: 16 pages + appendices. arXiv admin note: text overlap with arXiv:2106.01665

For a system of bosons that interact through a class of general memory kernels, a recurrence relation for the partition function is derived within the path-integral formalism. This approach provides a generalization to previously known treatments in the literature of harmonically coupled systems of identical particles. As an example the result is applied to the specific heat of a simplified model of an open quantum system of bosons, harmonically coupled to a reservoir of distinguishable fictitious masses., Comment: 19 pages, 3 figures

There is indeed a finite-size issue in computer simulations of the correlation decay in a quasi-one-dimensional system of hard disks. The correlation function obtained from the exact canonical $NVT$ partition function shows a decay which is much slower and more complex than the exponential one, presented in the Comment.

Recently we explained that the classical $Q$ Schur functions stand behind various well-known properties of the cubic Kontsevich model, and the next step is to ask what happens in this approach to the generalized Kontsevich model (GKM) with monomial potential $X^{n+1}$. We propose to use the Hall-Littlewood polynomials at the parameter equal to the $n$-th root of unity as a generalization of the $Q$ Schur functions from $n=2$ to arbitrary $n>2$. They are associated with $n$-strict Young diagrams and are independent of time-variables $p_{kn}$ with numbers divisible by $n$. These are exactly the properties possessed by the generalized Kontsevich model (GKM), thus its partition function can be expanded in such functions $Q^{(n)}$. However, the coefficients of this expansion remain to be properly identified. At this moment, we have not found any "superintegrability" property $\,\sim character$, which expressed these coefficients through the values of $Q$ at delta-loci in the $n=2$ case. This is not a big surprise, because for $n>2$ our suggested $Q$ functions are not looking associated with characters., Comment: 13 pages

We construct a map between a class of codes over F4 and a family of non-rational Narain CFTs. This construction is complementary to a recently introduced relation between quantum stabilizer codes and a class of rational Narain theories. From the modular bootstrap point of view we formulate a polynomial ansatz for the partition function which reduces modular invariance to a handful of algebraic easy-to-solve constraints. For certain small values of central charge our construction yields optimal theories, i.e. those with the largest value of the spectral gap.

In this work, we present a method to locate the dominant zero of the Energy Probability Distribution (EPD) Zeros method applied to the study of phase transitions. The technique strongly reduces computer processing time and also increases the accuracy of the results. As an example, we apply it to the 2D Ising model, comparing with both the exact Onsager's results and the previous EPD zeros method results. We show that, for large lattices, the processing time is drastically reduced when compared to other EPD zeros search procedures, whereas for small lattices the gain in accuracy allows very accurate predictions in the thermodynamic limit., Comment: 10 pages, 5 figures. Revised extended version

Within the recently developed Hamiltonian formulation of the Zimm and Bragg model we re-evaluate several size dependent approximations of model partition function. Our size analysis is based on the comparison of chain length N with the maximal correlation (persistence) length ξ of helical conformation. For the first time we re-derive the partition function of zipper model by taking the limits of the Zimm–Bragg eigenvalues. The critical consideration of applicability boundaries for the single-sequence (zipper) and the long chain approximations has shown a gap in description for the range of experimentally relevant chain lengths of 5–10 persistence lengths ξ. Correction to the helicity degree expression is reported. For the exact partition function we have additionally found, that: at N/ξ≈10 the transition temperature Tm reaches its asymptotic behavior of infinite N, the transition interval ΔT needs about a thousand persistence lengths to saturate at its asymptotic, infinite length value. Obtained results not only contribute to the development of the Zimm–Bragg model, but are also relevant for a wide range of Biotechnologies, including the Biosensing applications.

We consider the Dubrovin--Frobenius manifold of rank $2$ whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the enumeration of maps on surfaces, ribbon graphs, Grothendieck's dessins d'enfants, strictly monotone Hurwitz numbers, or lattice points in the moduli spaces of curves. Liu, Zhang, and Zhou conjectured that the full partition function of this Dubrovin--Frobenius manifold is a tau-function of the extended nonlinear Schr\"odinger hierarchy, an extension of a particular rational reduction of the Kadomtsev--Petviashvili hierarchy. We prove a version of their conjecture specializing the Givental--Milanov method that allows to construct the Hirota quadratic equations for the partition function, and then deriving from them the Lax representation., Comment: 52 pages, no figures

We establish the average-case hardness of the algorithmic problem of exact computation of the partition function associated with the Sherrington–Kirkpatrick model of spin glasses with Gaussian couplings and random external field. In particular, we establish that unless P=#P, there does not exist a polynomial-time algorithm to exactly compute the partition function on average. This is done by showing that if there exists a polynomial time algorithm, which exactly computes the partition function for inverse polynomial fraction (1/nO(1)) of all inputs, then there is a polynomial time algorithm, which exactly computes the partition function for all inputs, with high probability, yielding P=#P. The computational model that we adopt is finite-precision arithmetic, where the algorithmic inputs are truncated first to a certain level N of digital precision. The ingredients of our proof include the random and downward self-reducibility of the partition function with random external field; an argument of Cai et al. (In STACS 99 (Trier) (1999) 90–99 Springer) for establishing the average-case hardness of computing the permanent of a matrix; a list-decoding algorithm of Sudan (In 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996) (1996) 164–172 IEEE Comput. Soc. Press), for reconstructing polynomials intersecting a given list of numbers at sufficiently many points; and near-uniformity of the log-normal distribution, modulo a large prime p. To the best of our knowledge, our result is the first one establishing a provable hardness of a model arising in the field of spin glasses. Furthermore, we extend our result to the same problem under a different real-valued computational model, for example, using a Blum–Shub–Smale machine (In [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science (1988) 387–397 IEEE) operating over real-valued inputs. We establish that, if there exists a polynomial time algorithm which exactly computes the partition function for 34+1nO(1) fraction of all inputs, then there exists a polynomial time algorithm, which exactly computes the partition function for all inputs, with high probability, yielding P=#P. Our proof uses the random self-reducibility of the partition function, together with a control over the total variation distance for log-normal random variables in presence of a convex perturbation, and the Berlekamp–Welch algorithm.

The seminal Lee-Yang theorem states that for any graph the zeros of the partition function of the ferromagnetic Ising model lie on the unit circle in $\mathbb C$. In fact the union of the zeros of all graphs is dense on the unit circle. In this paper we study the location of the zeros for the class of graphs of bounded maximum degree $d\geq 3$, both in the ferromagnetic and the anti-ferromagnetic case. We determine the location exactly as a function of the inverse temperature and the degree $d$. An important step in our approach is to translate to the setting of complex dynamics and analyze a dynamical system that is naturally associated to the partition function., 24 pages, 3 figures. Made a number of small clarifications, corrections and changes in notation. Results remain unchanged. To appear in the Journal of the London Mathematical Society

MacMahon showed that the generating function for partitions into at most $k$ parts can be decomposed into a partial fractions-type sum indexed by the partitions of $k$. In this present work, a generalization of MacMahon's result is given, which in turn provides a full combinatorial explanation., 12 pages. Full account of the author's plenary talk at Combinatory Analysis 2018: in Honor of George Andrews' 80th Birthday, June 24, 2018

We prove several identities of the type ?$\alpha (n) = \Sigma_{k=0}^\infty \beta (\frac{(n - k(k + 1)/2)} {2})$?. Here, the functions ?$\alpha (n)$? and ?$\beta (n)$? count partitions with certain restrictions or the number of parts in certain partitions. Since G. N. Watson Proc. Lond. Math. Soc. (2) 42, 550-556 (1937) proved the identity for ?$\alpha (n) = Q(n)$?, the number of partitions of ?$n$? into distinct parts, and ?$\beta (n) = p(n)$?, Euler's partition function, we refer to these identities as Watson type identities. Our work is motivated by results of G. E. Andrews and M. Merca ''On the number of even parts in all partitions of $n$ into distinct parts'', Ann. Comb. (to appear) who recently discovered and proved new Euler type identities. We provide analytic proofs and explain how one could construct bijective proofs of our results. Dokažemo več identitet tipa ?$\alpha (n) = \Sigma_{k=0}^\infty \beta (\frac{(n - k(k + 1)/2)} {2})$?. Tukaj funkciji ?$\alpha (n)$? in ?$\beta (n)$? štejeta razčlenitve z določenimi omejitvami ali število delov v določenih razčlenitvah. Ker je Watson dokazal identiteto za ?$\alpha (n) = Q(n)$?, kjer je ?$Q(n)$? število razčlenitev števila ?$n$? na same različne dele, in za ?$\beta (n) = p(n)$?, kjer je ?$p(n)$? Eulerjeva razčlenitvena funkcija, tovrstne identitete imenujemo identitete Watsonovega tipa. Najino delo je motivirano z rezultati G. E. Andrewsa in drugega avtorja, ki je nedavno odkril in dokazal nove identitete Eulerjevega tipa. Podava analitične dokaze in razloživa, kako konstruirati bijektivne dokaze najinih rezultatov.

The Turan inequality and its higher order analog arise in the study of Maclaurin coefficients of an entire function in the Laguerre-Polya class. It is well known that if a real entire function ψ ( x ) is in the LP class, the Maclaurin coefficients satisfy both the Turan inequality and the higher order Turan inequality. Chen, Jia and Wang proved that for n ≥ 95 , the higher order Turan inequality holds for the partition function p ( n ) and the 3-rd associated Jensen polynomials p ( n ) + 3 p ( n + 1 ) x + 3 p ( n + 2 ) x 2 + p ( n + 3 ) x 3 have only real zeros. Recently, Griffin, Ono, Rolen and Zagier showed that Jensen polynomials for a large family of functions, including those associated to ξ ( s ) and the partition function, are hyperbolic for sufficiently large n. This result gave evidence for Riemann hypothesis. In this paper, we give a unified approach to investigate the higher order Turan inequality for the sequences { a n / n ! } n ≥ 0 , where a n satisfy a three-term recurrence relation. In particular, we prove higher order Turan inequality for the sequences { a n / n ! } n ≥ 0 , where a n are the Motzkin numbers, the Fine number, the Franel numbers of order 3 and the Domb numbers. As a consequence, for these combinatorial sequences, the 3-rd associated Jensen polynomials a n n ! + 3 a n + 1 ( n + 1 ) ! x + 3 a n + 2 ( n + 2 ) ! x 2 + a n + 3 ( n + 3 ) ! x 3 have only real zeros. Furthermore, for these combinatorial sequences we conjecture that for any given integer m ≥ 4 , there exists an integer N ( m ) such that for n > N ( m ) , the m-th associated Jensen polynomials ∑ i = 0 m ( m i ) a n + i ( n + i ) ! x i have only real zeros.

We study the large $N$ 't Hooft expansion of the chiral partition function of 2d $U(N)$ Yang-Mills theory on a torus. There is a long-standing puzzle that no explicit holomorphic anomaly equation is known for the partition function, although it admits a topological string interpretation. Based on the chiral boson interpretation we clarify how holomorphic anomaly arises and propose a natural anti-holomorphic deformation of the partition function. Our deformed partition function obeys a fairly traditional holomorphic anomaly equation. Moreover, we find a closed analytic expression for the deformed partition function. We also study the behavior of the deformed partition function both in the strong coupling/large area limit and in the weak coupling/small area limit. In particular, we observe that drastic simplification occurs in the weak coupling/small area limit, giving another nontrivial support for our anti-holomorphic deformation., Comment: 30 pages, 7 figures, v2: typos corrected, published version

Computing the partition function of a graphical model is a fundamental task in probabilistic inference. Variational bounds and Monte Carlo methods, two important approximate paradigms for this task, each has its respective strengths for solving different types of problems, but it is often nontrivial to decide which one to apply to a particular problem instance without significant prior knowledge and a high level of expertise. In this paper, we propose a general framework that interleaves optimization of variational bounds (via message passing) with Monte Carlo sampling. Our adaptive interleaving policy can automatically balance the computational effort between these two schemes in an instance-dependent way, which provides our framework with the strengths of both schemes, leads to tighter anytime bounds and an unbiased estimate of the partition function, and allows flexible tradeoffs between memory, time, and solution quality. We verify our approach empirically on real-world problems taken from recent UAI inference competitions.

We introduce the $q$-Hahn PushTASEP --- an integrable stochastic interacting particle system which is a 3-parameter generalization of the PushTASEP, a well-known close relative of the TASEP (Totally Asymmetric Simple Exclusion Process). The transition probabilities in the $q$-Hahn PushTASEP are expressed through the $_4\phi_3$ basic hypergeometric function. Under suitable limits, the $q$-Hahn PushTASEP degenerates to all known integrable (1+1)-dimensional stochastic systems with a pushing mechanism. One can thus view our new system as a pushing counterpart of the $q$-Hahn TASEP introduced by Povolotsky (2013). We establish Markov duality relations and contour integral formulas for the $q$-Hahn PushTASEP. In a $q\to 1$ limit of our process we arrive at a random recursion which, in a special case, appears to be similar to the inverse-Beta polymer model. However, unlike in recursions for Beta polymer models, the weights (i.e., the coefficients of the recursion) in our model depend on the previous values of the partition function in a nontrivial manner., Comment: 29 pages, 3 figures; v3: minor corrections and improvements of the presentation. To appear in IMRN