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Автор: Grafiati

Опубліковано: 14 березня 2023

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1

PUTROV, PAVEL, and MASAHITO YAMAZAKI. "EXACT ABJM PARTITION FUNCTION FROM TBA." Modern Physics Letters A 27, no. 34 (November 2, 2012): 1250200. http://dx.doi.org/10.1142/s0217732312502008.

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Анотація:

We report on the exact computation of the S3 partition function of U (N)k × U (N)-k ABJM theory for k = 1, N = 1, …, 19. The result is a polynomial in π-1 with rational coefficients. As an application of our results, we numerically determine the coefficient of the membrane 1-instanton correction to the partition function.

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2

Hatsuda, Yasuyuki, Sanefumi Moriyama, and Kazumi Okuyama. "Exact instanton expansion of the ABJM partition function." Progress of Theoretical and Experimental Physics 2015, no. 11 (October 28, 2015): 11B104. http://dx.doi.org/10.1093/ptep/ptv145.

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3

Izmailov, Alexander F., and Alexander R. Kessel. "Exact quantum partition function of the BCS model." International Journal of Theoretical Physics 29, no. 10 (October 1990): 1073–90. http://dx.doi.org/10.1007/bf00672086.

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4

Bouttier, J., P. Di Francesco, and E. Guitter. "Random trees between two walls: exact partition function." Journal of Physics A: Mathematical and General 36, no. 50 (December 1, 2003): 12349–66. http://dx.doi.org/10.1088/0305-4470/36/50/001.

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5

HATZINIKITAS, AGAPITOS, and IOANNIS SMYRNAKIS. "CLOSED BOSONIC STRING PARTITION FUNCTION IN TIME INDEPENDENT EXACT pp-WAVE BACKGROUND." International Journal of Modern Physics A 21, no. 05 (February 20, 2006): 995–1013. http://dx.doi.org/10.1142/s0217751x06025493.

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The modular invariance of the one-loop partition function of the closed bosonic string in four dimensions in the presence of certain homogeneous exact pp -wave backgrounds is studied. In the absence of an axion field, the partition function is found to be modular invariant and equal to the free field partition function. The partition function remains unchanged also in the presence of a fixed axion field. However, in this case, the covariant form of the action suggests summation over all possible twists generated by the axion field. This is shown to modify the partition function. In the light-cone gauge, the axion field generates twists only in the worldsheet σ-direction, so the resulting partition function is not modular invariant, hence wrong. To obtain the correct partition function one needs to sum over twists in the t-direction as well, as suggested by the covariant form of the action away from the light-cone gauge.

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6

Julian Lee. "Exact Partition Function Zeros of Two-Dimensional Lattice Polymers." Journal of the Korean Physical Society 44, no. 3 (March 15, 2004): 617. http://dx.doi.org/10.3938/jkps.44.617.

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7

Basu-Mallick, B., and Nilanjan Bondyopadhaya. "Exact partition function of supersymmetric Haldane–Shastry spin chain." Nuclear Physics B 757, no. 3 (November 2006): 280–302. http://dx.doi.org/10.1016/j.nuclphysb.2006.09.009.

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8

Zhang, Degang. "Exact Solution for Three-Dimensional Ising Model." Symmetry 13, no. 10 (October 1, 2021): 1837. http://dx.doi.org/10.3390/sym13101837.

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Анотація:

The three-dimensional Ising model in a zero external field is exactly solved by operator algebras, similar to the Onsager’s approach in two dimensions. The partition function of the simple cubic crystal imposed by the periodic boundary condition along two directions and the screw boundary condition along the third direction is calculated rigorously. In the thermodynamic limit an integral replaces a sum in the formula of the partition function. The critical temperatures, at which order–disorder transitions in the infinite crystal occur along three axis directions, are determined. The analytical expressions for the internal energy and the specific heat are also presented.

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9

González, Gabriel. "Exact Partition Function for the Random Walk of an Electrostatic Field." Advances in Mathematical Physics 2017 (2017): 1–5. http://dx.doi.org/10.1155/2017/6970870.

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The partition function for the random walk of an electrostatic field produced by several static parallel infinite charged planes in which the charge distribution could be either ±σ is obtained. We find the electrostatic energy of the system and show that it can be analyzed through generalized Dyck paths. The relation between the electrostatic field and generalized Dyck paths allows us to sum overall possible electrostatic field configurations and is used for obtaining the partition function of the system. We illustrate our results with one example.

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10

Kogan, Yaakov. "Exact analysis for a class of simple, circuit-switched networks with blocking." Advances in Applied Probability 21, no. 4 (December 1989): 952–55. http://dx.doi.org/10.2307/1427782.

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Анотація:

We consider the same circuit switching problem as in Mitra [1]. The calculation of the blocking probabilities is reduced to finding the partition function for a closed exponential pseudo-network with L−1 customers. This pseudo-network differs from that in [1] in one respect only: service rates at nodes 1, 2, …, p depend on the queue length. The asymptotic expansion developed in [1] follows from our exact expression for the partition function.

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11

Kogan, Yaakov. "Exact analysis for a class of simple, circuit-switched networks with blocking." Advances in Applied Probability 21, no. 04 (December 1989): 952–55. http://dx.doi.org/10.1017/s0001867800019200.

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Анотація:

We consider the same circuit switching problem as in Mitra [1]. The calculation of the blocking probabilities is reduced to finding the partition function for a closed exponential pseudo-network with L−1 customers. This pseudo-network differs from that in [1] in one respect only: service rates at nodes 1, 2, …, p depend on the queue length. The asymptotic expansion developed in [1] follows from our exact expression for the partition function.

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12

Skorik, Sergei. "Exact nonequilibrium current from the partition function for impurity-transport problems." Physical Review B 57, no. 20 (May 15, 1998): 12772–80. http://dx.doi.org/10.1103/physrevb.57.12772.

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13

Chang, Shu-Chiuan, and Robert Shrock. "Exact Potts model partition function on strips of the triangular lattice." Physica A: Statistical Mechanics and its Applications 286, no. 1-2 (October 2000): 189–238. http://dx.doi.org/10.1016/s0378-4371(00)00225-9.

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14

Iqbal, Amer, Babar A. Qureshi, and Khurram Shabbir. "(q, t) identities and vertex operators." Modern Physics Letters A 31, no. 11 (April 10, 2016): 1650065. http://dx.doi.org/10.1142/s0217732316500656.

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Анотація:

Using vertex operators acting on fermionic Fock space we prove certain identities, which depend on a number of parameters, generalizing and refining the Nekrasov–Okounkov identity. These identities provide exact product representation for the instanton partition function of certain five-dimensional quiver gauge theories. This product representation also clearly displays the modular transformation properties of the gauge theory partition function.

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15

Andriushchenko, Petr Dmitrievich, and Konstantin V. Nefedev. "Partition Function and Density of States in Models of a Finite Number of Ising Spins with Direct Exchange between the Minimum and Maximum Number of Nearest Neighbors." Solid State Phenomena 247 (March 2016): 142–47. http://dx.doi.org/10.4028/www.scientific.net/ssp.247.142.

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Анотація:

The results of studies of 1D Ising models and Curie-Weiss models partition functions structure are presented in this work. Exact calculation of the partition function using the authors combinatorial approach for such system is discussed. The distribution of the energy levels degeneracy was calculated. Analytical solution for density of states of 1D Ising chain were obtained. Generating functions for these models were obtained. It was suggested that in Curie-Weiss model the transition to a low-energy state occurs without the formation of separation boundaries

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16

IZMAILOV, ALEXANDER F., and ALEXANDER R. KESSEL. "SOLUTION OF THE BCS MODEL." International Journal of Modern Physics A 04, no. 18 (November 10, 1989): 4991–5002. http://dx.doi.org/10.1142/s0217751x89002120.

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Анотація:

The exact calculation of the reduced BCS model quantum partition function in the region of temperatures T > Tc was carried out by the path integration method. The partition function demonstrates the critical behavior at some temperature Tc. It turns out that this temperature is larger than the critical temperature T'c obtained in the traditional theories which are valid in the temperature region T < T'c.

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17

Guerin, H. "Exact classical vibrational-rotational partition function for Lennard-Jones and Morse potentials." Journal of Physics B: Atomic, Molecular and Optical Physics 25, no. 8 (April 28, 1992): 1697–703. http://dx.doi.org/10.1088/0953-4075/25/8/006.

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18

Basu-Mallick, Bireswar, Hideaki Ujino, and Miki Wadati. "Exact Spectrum and Partition Function of SU(m|n) Supersymmetric Polychronakos Model." Journal of the Physical Society of Japan 68, no. 10 (October 1999): 3219–26. http://dx.doi.org/10.1143/jpsj.68.3219.

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19

Jian-wei, Pan, Zhang Yong-de, and G. G. Siu. "Exact Expressions of Energy Spectrum and Partition Function for Quantum Quadratic Systems." Chinese Physics Letters 14, no. 4 (April 1997): 241–44. http://dx.doi.org/10.1088/0256-307x/14/4/001.

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20

Zhelifonov, M. P., and B. S. Nikitin. "Exact solution for the partition function of a system of interacting fermions." Theoretical and Mathematical Physics 68, no. 1 (July 1986): 707–14. http://dx.doi.org/10.1007/bf01017800.

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21

Lee, Julian. "Exact Partition Function Zeros of the Wako-Saitô-Muñoz-Eaton Protein Model." Biophysical Journal 106, no. 2 (January 2014): 438a. http://dx.doi.org/10.1016/j.bpj.2013.11.2467.

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22

Izmailov, A. F., and A. R. Kessel. "Partition function of the BCS model in the nonregular phase: Exact results." Physica C: Superconductivity 168, no. 3-4 (June 1990): 450–56. http://dx.doi.org/10.1016/0921-4534(90)90539-q.

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23

Tsuzuki, Toshio. "Dynamic compensation theorem and exact partition function of a spin-boson system." Solid State Communications 74, no. 8 (May 1990): 743–46. http://dx.doi.org/10.1016/0038-1098(90)90927-4.

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24

Aguilar, A., and E. Braun. "Exact solution of a general two-dimensional Ising model: The partition function." Physica A: Statistical Mechanics and its Applications 170, no. 3 (January 1991): 643–62. http://dx.doi.org/10.1016/0378-4371(91)90011-z.

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25

DAMGAARD, P. H., and J. LACKI. "PARTITION FUNCTION ZEROS OF AN ISING SPIN GLASS." International Journal of Modern Physics C 06, no. 06 (December 1996): 819–43. http://dx.doi.org/10.1142/s012918319500068x.

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Анотація:

We study the pattern of zeros emerging from exact partition function evaluations of Ising spin glasses on conventional finite lattices of varying sizes. A large number of random bond configurations are probed in the framework of quenched averages. This study is motivated by the relationship between hierarchical lattice models whose partition function zeros fall on Julia sets and chaotic renormalization group flows in such models with frustration, and by the possible connection of the latter with spin glass behavior. In any finite volume, the simultaneous distribution of the zeros of all partition functions can be viewed as part of the more general problem of finding the location of all the zeros of a certain class of random polynomials with positive integer coefficients. Some aspects of this problem have been studied in various areas of mathematics, and we show in particular how polynomial mappings which are used in graph theory to classify graphs, may help in characterizing the distribution of zeros. We finally discuss the possible limiting set of these zeros as the volume is sent to infinity.

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26

CHANG, SHU-CHIUAN, and ROBERT SHROCK. "EXACT PARTITION FUNCTION FOR THE POTTS MODEL WITH NEXT-NEAREST NEIGHBOR COUPLINGS ON ARBITRARY-LENGTH LADDERS." International Journal of Modern Physics B 15, no. 05 (February 20, 2001): 443–78. http://dx.doi.org/10.1142/s0217979201004630.

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We present exact calculations of partition function Z of the q-state Potts model with next-nearest-neighbor spin–spin couplings, both for the ferromagnetic and antiferromagnetic case, for arbitrary temperature, on n-vertex ladders with free, cyclic, and Möbius longitudinal boundary conditions. The free energy is calculated exactly for the infinite-length limit of these strip graphs and the thermodynamics is discussed. Considering the full generalization to arbitrary complex q and temperature, we determine the singular locus ℬ in the corresponding [Formula: see text] space, arising as the accumulation set of partition function zeros as n → ∞.

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27

Archibald, Margaret, Aubrey Blecher, Charlotte Brennan, Arnold Knopfmacher, and Toufik Mansour. "Two by two squares in set partitions." Mathematica Slovaca 70, no. 1 (February 25, 2020): 29–40. http://dx.doi.org/10.1515/ms-2017-0328.

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AbstractA partition π of a set S is a collection B1, B2, …, Bk of non-empty disjoint subsets, alled blocks, of S such that $\begin{array}{} \displaystyle \bigcup _{i=1}^kB_i=S. \end{array}$ We assume that B1, B2, …, Bk are listed in canonical order; that is in increasing order of their minimal elements; so min B1 < min B2 < ⋯ < min Bk. A partition into k blocks can be represented by a word π = π1π2⋯πn, where for 1 ≤ j ≤ n, πj ∈ [k] and $\begin{array}{} \displaystyle \bigcup _{i=1}^n \{\pi_i\}=[k], \end{array}$ and πj indicates that j ∈ Bπj. The canonical representations of all set partitions of [n] are precisely the words π = π1π2⋯πn such that π1 = 1, and if i < j then the first occurrence of the letter i precedes the first occurrence of j. Such words are known as restricted growth functions. In this paper we find the number of squares of side two in the bargraph representation of the restricted growth functions of set partitions of [n]. These squares can overlap and their bases are not necessarily on the x-axis. We determine the generating function P(x, y, q) for the number of set partitions of [n] with exactly k blocks according to the number of squares of size two. From this we derive exact and asymptotic formulae for the mean number of two by two squares over all set partitions of [n].

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28

LI, Y. M., Y. YU, N. D'AMBRUMENIL, L. YU, and Z. B. SU. "EXPRESSION OF THE PARTITION FUNCTION AND RENORMALIZATION EQUATION OF SPIN-1/2 TOMONAGA–LUTTINGER CHAINS." Modern Physics Letters B 08, no. 12 (May 20, 1994): 749–57. http://dx.doi.org/10.1142/s0217984994000753.

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Анотація:

We derive an exact expression for the partition function of two spin-1/2 Tomonaga–Luttinger chains, and obtain the renormalization group equation for the normal state. Anderson's confinement is then discussed. We find that spin–charge separation is irrelevant to the confinement.

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29

Chang, Shu-Chiuan, and Robert Shrock. "Some exact results on the Potts model partition function in a magnetic field." Journal of Physics A: Mathematical and Theoretical 42, no. 38 (September 7, 2009): 385004. http://dx.doi.org/10.1088/1751-8113/42/38/385004.

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30

Wood, D. W. "The exact location of partition function zeros, a new method for statistical mechanics." Journal of Physics A: Mathematical and General 18, no. 15 (October 21, 1985): L917—L921. http://dx.doi.org/10.1088/0305-4470/18/15/003.

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31

Lee, Jae Hwan, Seung-Yeon Kim, and Julian Lee. "Parallel algorithm for calculation of the exact partition function of a lattice polymer." Computer Physics Communications 182, no. 4 (April 2011): 1027–33. http://dx.doi.org/10.1016/j.cpc.2011.01.004.

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32

LU, HUAI-XIN, ZENG-BING CHEN, LEI MA, and YONG-DE ZHANG. "EXACT SOLUTION FOR A GENERAL SUPERSYMMETRIC QUADRATIC SYSTEM." Modern Physics Letters B 16, no. 07 (March 20, 2002): 241–50. http://dx.doi.org/10.1142/s0217984902003671.

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We construct a general supersymmetric quantum transformation for studying the supersymmetric quadratic Hamiltonian which contains the interaction of multi-mode fermions with multi-mode bosons. The Hamiltonian can be derived from that of multi-mode radiation field interacting with m two-level atoms. By using the supersymmetric quantum transformation, the diagonolized Hamiltonian is given. We also concisely derive the analytic expression of the super-partition function for the supersymmetric quadratic Hamiltonian without knowing the energy spectrum.

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33

LIN, K. Y. "EXACT RESULTS FOR THE ISING MODEL ON A 3–12 LATTICE." International Journal of Modern Physics B 03, no. 08 (August 1989): 1237–45. http://dx.doi.org/10.1142/s021797928900083x.

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We consider the Ising model on a 3–12 lattice with magnetic field. An exact functional relation is established for the partition function and our result is a generalization of Giacomini’s work on the Kagomé lattice. We calculate the zero-field magnetic susceptibility when an appropriate relation among the interaction parameters is satisfied.

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34

Saryal, Sushant, and Deepak Dhar. "Exact results for interacting hard rigid rotors on a d-dimensional lattice." Journal of Statistical Mechanics: Theory and Experiment 2022, no. 4 (April 1, 2022): 043204. http://dx.doi.org/10.1088/1742-5468/ac6038.

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Abstract We study the entropy of a set of identical hard objects, of general shape, with each object pivoted on the vertices of a d-dimensional regular lattice of lattice spacing a, but can have arbitrary orientations. When the pivoting point is situated asymmetrically on the object, we show that there is a range of lattice spacings a, where in any orientation, a particle can overlap with at most one of its neighbors. In this range, the entropy of the system of particles can be expressed exactly in terms of the grand partition function of coverings of the base lattice by dimers at a finite negative activity. The exact entropy in this range is fully determined by the second virial coefficient. Calculation of the partition function is also shown to be reducible to that of the same model with discretized orientations. We determine the exact functional form of the probability distribution function of orientations at a site. This depends on the density of dimers for the given activity in the dimer problem, which we determine by summing the corresponding Mayer series numerically. These results are verified by numerical simulations.

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35

CHATTERJEE, R. "EXACT PARTITION FUNCTION AND BOUNDARY STATE OF CRITICAL ISING MODEL WITH BOUNDARY MAGNETIC FIELD." Modern Physics Letters A 10, no. 12 (April 20, 1995): 973–84. http://dx.doi.org/10.1142/s0217732395001071.

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Анотація:

We compute the exact partition function of the 2-D Ising Model at critical temperature but with nonzero magnetic field at the boundary. The model describes a renormalization group flow between the free and fixed conformal boundary conditions in the space of boundary interactions. For this flow the universal ground state degeneracy g and the full boundary state is computed exactly.

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36

Savvidy, George. "The gonihedric paradigm extension of the Ising model." Modern Physics Letters B 29, no. 32 (November 30, 2015): 1550203. http://dx.doi.org/10.1142/s0217984915502036.

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In this paper we review a recently suggested generalization of the Feynman path integral to an integral over random surfaces. The proposed action is proportional to the linear size of the random surfaces and is called gonihedric. The convergence and the properties of the partition function are analyzed. The model can also be formulated as a spin system with identical partition functions. The spin system represents a generalization of the Ising model with ferromagnetic, antiferromagnetic and quartic interactions. Higher symmetry of the model allows to construct dual spin systems in three and four dimensions. In three dimensions the transfer matrix describes the propagation of closed loops and we found its exact spectrum. It is a unique exact solution of the three-dimensional statistical spin system. In three and four dimensions, the system exhibits the second-order phase transitions. The gonihedric spin systems have exponentially degenerated vacuum states separated by the potential barriers and can be used as a storage of binary information.

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37

Verbaarschot, J. J. M., and T. Wettig. "Random Matrix Theory and Chiral Symmetry in QCD." Annual Review of Nuclear and Particle Science 50, no. 1 (December 2000): 343–410. http://dx.doi.org/10.1146/annurev.nucl.50.1.343.

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Анотація:

▪ Abstract Random matrix theory is a powerful way to describe universal correlations of eigenvalues of complex systems. It also may serve as a schematic model for disorder in quantum systems. In this review, we discuss both types of applications of chiral random matrix theory to the QCD partition function. We show that constraints imposed by chiral symmetry and its spontaneous breaking determine the structure of low-energy effective partition functions for the Dirac spectrum. We thus derive exact results for the low-lying eigenvalues of the QCD Dirac operator. We argue that the statistical properties of these eigenvalues are universal and can be described by a random matrix theory with the global symmetries of the QCD partition function. The total number of such eigenvalues increases with the square root of the Euclidean four-volume. The spectral density for larger eigenvalues (but still well below a typical hadronic mass scale) also follows from the same low-energy effective partition function. The validity of the random matrix approach has been confirmed by many lattice QCD simulations in a wide parameter range. Stimulated by the success of the chiral random matrix theory in the description of universal properties of the Dirac eigenvalues, the random matrix model is extended to nonzero temperature and chemical potential. In this way we obtain qualitative results for the QCD phase diagram and the spectrum of the QCD Dirac operator. We discuss the nature of the quenched approximation and analyze quenched Dirac spectra at nonzero baryon density in terms of an effective partition function. Relations with other fields are also discussed.

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38

Lee, Julian, and Koo-Chul Lee. "Exact zeros of the partition function for a continuum system with double Gaussian peaks." Physical Review E 62, no. 4 (October 1, 2000): 4558–63. http://dx.doi.org/10.1103/physreve.62.4558.

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39

Lee, Jae Hwan, Seung-Yeon Kim, and Julian Lee. "Exact partition function zeros and the collapse transition of a two-dimensional lattice polymer." Journal of Chemical Physics 133, no. 11 (September 21, 2010): 114106. http://dx.doi.org/10.1063/1.3486176.

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40

Kouzoudis, D. "Exact analytical partition function and spin gap for a 2×3 quantum spin ladder." Journal of Magnetism and Magnetic Materials 214, no. 1-2 (May 2000): 112–18. http://dx.doi.org/10.1016/s0304-8853(99)00809-4.

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41

Lee, Julian. "Transfer matrix algorithm for computing the exact partition function of a square lattice polymer." Computer Physics Communications 228 (July 2018): 11–21. http://dx.doi.org/10.1016/j.cpc.2018.03.022.

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42

IGUCHI, KAZUMOTO, and KAZUHIKO AOMOTO. "INTEGRAL REPRESENTATION FOR THE GR AND PARTITION FUNCTION IN QUANTUM STATISTICAL MECHANICS OF EXCLUSION STATISTICS." International Journal of Modern Physics B 14, no. 05 (February 20, 2000): 485–506. http://dx.doi.org/10.1142/s0217979200000455.

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Анотація:

We derive an exact integral representation for the gr and partition function for an ideal gas with exclusion statistics. Using this we show how the Wu's equation for the exclusion statistics appears in the problem. This can be an alternative proof for the Wu's equation. We also discuss that singularities are related to the existence of a phase transition of the system.

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43

Tamm, Mikhail V., Maxym Dudka, Nikita Pospelov, Gleb Oshanin, and Sergei Nechaev. "From steady-state TASEP model with open boundaries to 1D Ising model at negative fugacity." Journal of Statistical Mechanics: Theory and Experiment 2022, no. 3 (March 1, 2022): 033201. http://dx.doi.org/10.1088/1742-5468/ac52a5.

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Abstract We expose a series of exact mappings between particular cases of four statistical physics models: (i) equilibrium 1D lattice gas with nearest-neighbor repulsion, (ii) (1 + 1)D combinatorial heap of pieces, (iii) directed random walks on a half-plane, and (iv) 1D totally asymmetric simple exclusion process (TASEP). In particular, we show that generating function of a 1D steady-state TASEP with open boundaries can be interpreted as a quotient of partition functions of 1D hard-core lattice gases with one adsorbing lattice site and negative fugacity. This result is based on the combination of a representation of a steady-state TASEP configurations in terms of (1 + 1)D heaps of pieces (HP) and a theorem of X Viennot which projects the partition function of (1 + 1)D HP onto that of a single layer of pieces, which in this case is a 1D hard-core lattice gas.

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44

HANSEL, D., and J. M. MAILLARD. "FORMAL CONSTRAINTS ON SERIES ANALYSIS ON THE POTTS MODEL." Modern Physics Letters B 01, no. 04 (July 1987): 145–53. http://dx.doi.org/10.1142/s021798498700020x.

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It is shown that the low temperature expansion of the partition function, magnetization and nearest neighbour correlation functions of the q-state checkerboard Potts model in a magnetic field drastically simplify on a very simple algebraic variety. These four formal constraints on the expansions are also analysed in the framework of the resummed low temperature expansions and the large q expansions. These exact results are generalized straightforwardly to higher dimensional hypercubic lattices and also to some random problems.

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45

Badasyan, Artem. "System Size Dependence in the Zimm–Bragg Model: Partition Function Limits, Transition Temperature and Interval." Polymers 13, no. 12 (June 17, 2021): 1985. http://dx.doi.org/10.3390/polym13121985.

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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.

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46

Oi, Takao. "Ab initio Molecular Orbital Calculations of Reduced Partition Function Ratios of Polyboric Acids and Polyborate Anions." Zeitschrift für Naturforschung A 55, no. 6-7 (July 1, 2000): 623–28. http://dx.doi.org/10.1515/zna-2000-6-710.

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Abstract Molecular orbital calculations at the HF/6-3 lG(d) level were carried out for polyboric acids and poly-borate anions up to a pentamer to estimate their 11B -to-10B isotopic reduced partition function ratios (RPFRs) and examine the additivity of logarithms of RPFRs. Approximate RPFR-values calculated by the use of the additivity agreed with exact RPFR-values within a margin of 1% error. This error was equivalent to a 5% error on ln(RPFR). The equilibrium constants of mono boron isotope exhange reac-tions between three-coordinate boron and four-coordinate boron ranged from 1.0203 to 1.0360 at 25 °C, indicating the importance of exact evaluation of RPFRs of polymers.

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47

Penniston, David. "11-Regular partitions and a Hecke eigenform." International Journal of Number Theory 15, no. 06 (July 2019): 1251–59. http://dx.doi.org/10.1142/s1793042119500696.

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A partition of a positive integer [Formula: see text] is called [Formula: see text]-regular if none of its parts is divisible by [Formula: see text]. Let [Formula: see text] denote the number of 11-regular partitions of [Formula: see text]. In this paper we give a complete description of the behavior of [Formula: see text] modulo [Formula: see text] when [Formula: see text] in terms of the arithmetic of the ring [Formula: see text]. This description is obtained by relating the generating function for these values of [Formula: see text] to a Hecke eigenform, and as a byproduct we find exact criteria for which of these values are divisible by 5 in terms of the prime factorization of [Formula: see text].

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48

Perfilieva, Irina, Tam Pham, and Petr Ferbas. "Quadrature Rules for the Fm-Transform Polynomial Components." Axioms 11, no. 10 (September 25, 2022): 501. http://dx.doi.org/10.3390/axioms11100501.

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The purpose of this paper is to reduce the complexity of computing the components of the integral Fm-transform, m≥0, whose analytic expressions include definite integrals. We propose to use nontrivial quadrature rules with nonuniformly distributed integration points instead of the widely used Newton–Cotes formulas. As the weight function that determines orthogonality, we choose the generating function of the fuzzy partition associated with the Fm-transform. Taking into account this fact and the fact of exact integration of orthogonal polynomials, we obtain exact analytic expressions for the denominators of the components of the Fm-transformation and their approximate analytic expressions, which include only elementary arithmetic operations. This allows us to effectively estimate the components of the Fm-transformation for 0≤m≤3. As a side result, we obtain a new method of numerical integration, which can be recommended not only for continuous functions, but also for strongly oscillating functions. The advantage of the proposed calculation method is shown by examples.

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49

Karandashev, Ya M., and M. Yu Malsagov. "Polynomial algorithm for exact calculation of partition function for binary spin model on planar graphs." Optical Memory and Neural Networks 26, no. 2 (April 2017): 87–95. http://dx.doi.org/10.3103/s1060992x17020035.

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50

Guttmann, A. J. "Comment on 'The exact location of partition function zeros, a new method for statistical mechanics'." Journal of Physics A: Mathematical and General 20, no. 2 (February 1, 1987): 511–12. http://dx.doi.org/10.1088/0305-4470/20/2/037.

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