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Fa Yueh (Fred) Wu was born on 5 January 1932 in Nanking (now known as Nanjing), China, the capital of the Nationalist government. Wu began kindergarten in 1937 in a comfortable setting, as his father held a relatively high government position. But the Sino–Japanese war broke out in July 1937, and Nanking fell to Japanese hands in November. Fleeing the Japanese, his parents brought Wu to their hometown in Hunan, and then to the war capital Chungking (now Chongqing) in 1938, where they lived for eight years until the end of the war.

Around that time the Japanese began bombing Chungking, and Wu's childhood memories were dominated by air raids, bombings and burning not dissimilar to those experienced by Londoners during the war. At times the air raids lasted for days disrupting everyday lives in Chungking, including Wu's schooling. One day during a fierce bombing raid, a bomb fell in their garden reducing a pavilion and the surrounding pond to a huge crater; another bomb fell just a few metres from the tunnel where his family took shelter, almost sealing the only entrance. The family moved the very next day to the countryside.

As a result of the war, Wu attended seven schools before finishing his primary education. Fortunately, by the time he entered junior high school in 1943, the Japanese forces were on the wane and Wu entered the elite middle school, Nankai. His early academic performance in Nankai seemed to him mediocre, but he nevertheless impressed his geometry teacher by showing bursts of talent. With hindsight, this early interest in geometry may have led to his later insights in graphical analyses of statistical systems. The family returned to Nanking in 1946 after the Victory over Japan Day. By this time his father had become elected to the Legislative Yuan, the equivalent of the US Senate.

Wu entered high school in Nanking in 1946. Since he came from an elite school in Chungking, he excelled in most of his classes, especially mathematics. Notwithstanding his academic success, Wu probably spent more time playing Chinese chess, a board game similar to international chess. He ranked high in a city-wide tournament and often played blind-folded games. He also spent time playing bridge, a game he learned in Nankai and kept up throughout his years in the US. He also loved puzzles and riddles.

But the good days did not last long, as the civil war drew closer to Nanking with the Communists winning. The family fled Nanking once again, following a zigzag path and traveling by boat, train, car and then by boat again, eventually reaching Taiwan in June 1949. By this time, the Nationalists had lost most of China, and there was no hope of returning to the mainland. Wu entered the Naval College of Technology to study electrical engineering, giving up an opportunity to study mathematics in the National Taiwan University, although his real interest was in mathematics.

In 1954, Wu graduated from the Naval College with a BS degree and the commission of Ensign. Recognizing his outstanding academic record in the College, the Chinese Navy sent him to the US in 1955 to study radar and sonar and to receive training as an instructor. He stayed at the Naval School of Electronics in San Francisco and at the Instructor's School in San Diego. Wu felt that he benefited from the instructor's training much more than from the electronics school, as the training helped him to develop teaching and presentation skills that served him well throughout his career. The Navy assigned him to teach electronics in the Naval Academy when he returned to Taiwan in 1956.

Wu was interested in attending a graduate school. The only institution that offered a graduate degree in Taiwan at the time was Taiwan's newly re-established Tsing Hua University. In its hurried retreat to Taiwan, the Nationalist government left the original Tsing Hua University, one of China's best-known institutions of higher learning with a history dating back to the 19th century, behind in Beijing. In 1956, after gaining footing in Taiwan, the Nationalist government revived Tsing Hua, and began offering a two-year Master's degree in nuclear science. Wu decided to apply for admission but faced considerable obstacles since he was in the Navy. After one year's effort, mostly on his father's part, Wu finally entered Tsing Hua in 1957. He completed the two-year program with an experimental thesis in 1959. By this time, the US was pushing for a scientific renaissance after the launch of the Soviet satellite Sputnik. Wu received offers of teaching assistantships from several physics departments in the US, and chose to continue his graduate education at Washington University in St. Louis in 1959.

At Washington University he studied many-body theory under the late Eugene Feenberg and produced several influential papers [1, 2] on ground state properties of liquid helium-3 and liquid helium-4. In 1963, he published a paper on formulating cluster expansions in an N-body problem [3], extending the Mayer expansion to systems with indexed many-body interactions, which appeared to have escaped the attention of the community of statistical physics that it deserved. The expansion made extensive use of graphical terms, demonstrating his prowess in graphical analysis at an early stage of his career. Wu's interest in many-body theory continued over the years, with subsequent works on the electron gas, adsorbed systems, and the long-perplexing problem of density correlations in Fermi and Bose systems. After obtaining his PhD from Washington University in 1963, Wu went on to teach at Virginia Polytechnic Institute (VPI) as an assistant professor.

In February 1967, Wu met Elliott Lieb who was visiting VPI to give a talk on the Bethe ansatz evaluation of the entropy of two-dimensional ice, a 6-vertex model. Wu soon realized the underlying graphical aspects of two-dimensional vertex models and solved the thermodynamics of a related 5-vertex model using the Pfaffian approach. The result was published in the April issue of Physical Review Letters (PRL) of the same year [4], and in September 1967, Wu moved to Northeastern University to join Lieb's group.

Wu taught at Northeastern for 39 years until his retirement in 2006 as the Matthews Distinguished University Professor of Physics. Over the years, Wu has published more than 230 papers and monographs, and he continues to publish after retirement. Most of his research since 1967 is in exact and rigorous analyses of lattice models and integrable systems, which is the theme of this special issue.

In 1968, after Wu's arrival at Northeastern, Lieb and Wu obtained the exact solution of the ground state of the one-dimensional Hubbard model and published the result in PRL [5], a work which has since become highly important after the advent of high-temperature superconductivity. This Lieb–Wu paper and Wu's 1982 review of the Potts model in Reviews of Modern Physics [37] are among the most cited papers in condensed matter physics. Later in 1968 Lieb departed Northeastern for MIT. As a result, the full version of the solution was not published until 34 years later [38] when Lieb and Wu collaborated to work on the manuscript on the occasion of Wu's 70th birthday.

Wu spent the summer of 1968 at Stony Brook as the guest of C N Yang. Working with Yang's student, C Fan, he extended the Pfaffian solution of the Ising model to general lattices and termed such models 'free-fermion', a term now in common use [6].

In 1972, Wu visited R J Baxter, whom he had met earlier in 1968 at MIT, in Canberra, Australia, with the support of a Fulbright grant. They solved the triangular-lattice Ising model with 3-spin interactions [7], a model now known as the Baxter–Wu model. It was an ideal collaboration. While Baxter derived the solution algebraically, Wu used graphical methods to reduce the problem to an Ashkin–Teller model, which greatly simplifies the presentation. While in Canberra, Wu also studied the 8-vertex model on the honeycomb lattice [8], a model which proved to be useful in his later research.

In 1973, Wu returned to Tsing Hua as a visiting professor and worked with colleague K Y Lin. They published two important papers introducing staggered vertex models for the first time [10, 11]. In other important work they clarified the nature of the phase diagram of the Ashkin–Teller model, and found it to have two phase transitions [9].

In the 1970s Wu traveled to Taiwan, Australia, Europe and to China when it re-opened. He met H N V Temperley in Aberdeen, Scotland in 1976, and collaborated with H J Brascamp and H Kunz in Lausanne to establish a number of rigorous results on vertex models, including a proof of the equivalence of boundary conditions for the 6-vertex model [13, 14]. From 1979 to 1980, Wu resided in the Netherlands and Germany, where he was the guest of Piet Kasteleyn at Leiden, Hans van Leeuwen at Delft and Kurt Binder in Juelich.

It was in Juelich that Wu completed the 1982 review paper on the Potts model [37], a paper that has been cited 70 or more times every year since its publication. Another important work in that period is a 1976 graphical analysis of the Potts model on the triangular lattice in collaboration with Baxter and Kelland [15]. This paper provided an elegant and conceptually easy description of the duality relation of the model, complementary to the algebraic analysis of Temperley and Lieb [16]. Four years later, Wu and Lin further refined the graphical aspects to reduce the model to a 5-vertex model, under which the duality relation appears as a simple spatial symmetry [18]. The Wu–Lin formulation of the Potts model is used by Jacobsen and Sculland in an analysis of the kagome-lattice Potts model in their first paper in this issue [39]. In other pioneering work in 1976, Wu and Y K Wang introduced a spin model with chiral interactions and its duality relation in Fourier space [19]. Prior to that time, studies of spin models had invariably been confined to models with symmetric interactions.

In 1977 Wu published an influential paper on spanning trees [20]. In it, he derived the spanning tree constants of the regular two-dimensional lattices. Since then, he has been the co-author of several papers extending this work to a variety of other two-dimensional Archimedean lattices [21–23]. In this issue Guttmann¹ and Rogers solve the three-dimensional version of this problem, which has resisted attack for more than 30 years [40]. The connection between spanning trees and dimers was previously highlighted by Neville Temperley in 1974 [17].

The ideas from number theory needed to obtain the spanning tree constant of three-dimensional lattices, notably logarithmic Mahler measures, are further discussed in the article by Glasser² in this issue [41].

Wu has had a long and productive collaboration with Maillard, particularly on aspects of the Ising model. Maillard also wrote the definitive description of Wu's many scientific contributions at the time of Wu's 70th birthday [24]. The paper was later included among the biographies of great names such as Newton and Feynman in the History of Physics: Individual Biographies section in the MIT Net Advance of Physics website [59]. Further developments in the Ising model are highlighted in the article by Boukraa, Hassani and Maillard³ in this issue [42]. Maillard's article also appears as the introduction to a wonderful collection of Wu's works that appeared in 2009 [25], entitled Exactly Solved Models: A Journey in Statistical Mechanics.

The relation between bond percolation and the random-cluster formulation of the Potts model was pioneered by Kasteleyn and Fortuin in 1969 [26]. Later, in a 1977 paper, Wu showed how to rederive this relation in a different setting and used it to obtain various quantities of interest in the bond percolation problem, including critical exponents, from the exact solution of the Potts model [27]. A few months later, in collaboration with Kunz, he showed that site percolation can also be related to the Potts model [28]. Problems in bond percolation are treated in this issue by several works. The paper by Hu, Blöte⁴ and Deng⁵ investigates how the imposition of a 'canonical' constraint, that there be an equal number of open and closed edges, affects the universal properties [43]. The paper by Ziff⁶, Scullard, Wierman and Sedlock exactly solves inhomogeneous percolation on lattices of the bow-tie and checkerboard types [44].

In a 1979 paper on Potts model critical points, Wu proposed a conjecture, now known as the homogeneity hypothesis, on the location of the critical point of the kagome lattice [29]. Since then, numerous studies have been carried out to test the validity of that conjecture [12]. However, many of these tests proved to be inconclusive since they produced results extremely close to the conjectured value. The puzzle is finally solved by Jacobsen and Scullard in their two papers in this issue [39, 45]. Using a graphical analysis based on the Wu–Lin 5-vertex formulation, they recover the Wu conjecture of the kagome-lattice critical point as the first-order approximation in a well-defined graphical analysis. This establishes once and for all the approximate nature of the Wu conjecture.

These investigations, and the exact solutions found by Wu, raised the question as to the conditions under which a lattice model is exactly solvable. Quite recently, such questions have been addressed through the technique of discrete holomorphicity (DH). This direction is represented in this issue by the contributions of Alam and Batchelor⁷, where the connection between DH and Yang–Baxter integrability is investigated [46]. DH is also a key ingredient in recent rigorous proofs that certain lattice models converge, in the continuum limit, to conformally invariant probabilistic processes known as Schramm–Loewner evolution (SLE). The theme of SLE appears within this issue in the article by Alberts, Kozdron and Lawler [47]. Finally, DH observables are used in this issue by Duminil-Copin to prove the divergence of the correlation length for the Potts model (in its formulation in terms of Fortuin–Kasteleyn clusters) when 1 ⩽ q ⩽ 4 [48].

Establishing the phase diagrams of lattice models is a recurrent theme in Wu's works. In an interesting but little-known work from 2000 with Guo and Blöte [30], he has shown that, contrary to common belief, the O(n) model on the honeycomb lattice has a second-order phase transition for n > 2. The question of phase diagrams for O(n)-type models is taken up in this issue by Blöte, Wang and Guo⁸ [49].

In 1983–84, Wu joined the National Science Foundation as the Director of the Condensed Matter Theory Program for 18 months. His duty was managing funding to individual researchers in condensed matter theory in the US. The 18-month tour in Washington offered Wu a bird's-eye view of condensed matter physics research in US universities, an understanding that proved useful to his later researches.

Throughout his career, Wu has insisted on the general applicability of graphical analysis to a variety of lattices. This aspect was highlighted in his 1988 paper on the Potts model and graph theory [31], in which he derived a number of equivalences with (di)chromatic and flow polynomials on arbitrary planar graphs, both for the partition function and correlation functions. An earlier result in the same vein is the equivalence of the Potts model on a planar graph with a loop model on the corresponding medial graph, found in 1976 in collaboration with Baxter and Kelland [15]. Building on these results, and on recent progress in the combinatorial approach to planar maps, Borot, Bouttier and Guitter systematically investigate properties of percolation and Potts models on random planar maps in their contribution to this issue [50].

Wu has published extensively on dimer enumerations. His work includes exact enumerations on non-orientable surfaces and surfaces with a single boundary defect. In this issue, Lu and Zhang consider dimer enumerations on the Klein bottle, which is an example of a non-orientable surface [51]. Another contribution is the paper by Ciucu and Fischer, considering dimer coverings of a domain with a defect (hole) in the interior [52].

Wu has also worked extensively in knot theory. He has constructed new knot invariants based on statistical mechanical models [61, 62], and published a well-received review of knot invariants for physicists [32], which elucidates the connection of knot invariants with statistical mechanics.

In 2004, Wu presented a new formulation of resistance networks [33], which permits the derivation of the exact expression of the resistance between two arbitrary nodes in any network. He later extended the formulation to impedance networks [34], a work which has since attracted interest in applications in petroleum research. These works can perhaps be seen as a distant echo of Wu's Navy training in electronics, more than 50 years earlier. In recent years Wu has developed this topic in joint work with Essam⁹, who together with Brak has related work on lattice paths in this issue [53]. A cognate paper by Arrowsmith, Bhatti and Essam also appears [54].

Wu has made other contributions to asymptotic analysis, for example in relation to dimers in his recent papers, where he also uses results from conformal invariance [60]. This thread is taken up by the article of Izmailian¹⁰ in this issue [55].

In 1997, Wu reported, in a short paper, a new formulation of duality relations of Potts correlation functions for n spins residing on the boundary of a lattice [35]. He gave the examples of n = 2 and 3, and remarked that the formulation can be extended to higher values of n 'in a straightforward fashion'. But the extension is by no means straightforward¹¹ and its solution was eventually found by Wu and his student H Y Huang [36]. They found that the correlation functions are not all independent when n = 4 and higher. They also deduced the connecting relations expressing crossing correlations in terms of non-crossing correlations, thus resolving the discrepancy.

Nowadays the interest in integrable systems largely transcends the realm of equilibrium statistical physics. Important and fundamental applications have appeared in out-of-equilibrium physics, in combinatorics, and in the study of certain dualities between string theories and gauge theories known under the common epithet of AdS/CFT duality. This last trend is represented in this issue by the contribution of Kostov [56].

Other interests of Wu in both quantum and classical systems are reflected in the article by Barry¹², Muttalib and Tanaka [57], and in the paper by Bauer, Bernard and Benoist on iterated stochastic measurements [58]. This latter paper appears very timely, since it is inspired by the experiments carried out in the group of Serge Haroche who earned the 2012 Nobel Prize in Physics.

Wu met his wife Ching Tse (Jane) in Taipei. They married in 1963 in St. Louis, Missouri. They have three daughters; Yvonne, a Professor of Child Neurology at the University of California San Francisco, Yolanda, a women's rights lawyer and a teacher of Suzuki violin, and Yelena, a postdoc in Child Clinical Psychology at Cincinnati Children's Hospital. Fred and Jane have five grandchildren. Wu left four siblings behind when he left China in 1949, and reunited with them after a 30-year separation for the first time in 1979. Two sisters and one brother are now deceased, and his younger brother, who also has three daughters, lives a comfortable life in retirement in Kunming, China.

It has been a pleasure to assemble this collection of papers on the occasion of Fred's 80th birthday, and we wish to thank him for providing much of the biographical information on which this introduction is based. We are also grateful to all the contributors for providing such a diverse and decidedly very modern panorama of the topic of lattice models and integrability, and for meeting the strict deadlines necessary to ensure the completion of this issue before the year 2012 draws to an end.

Fred Wu continues to be a highly productive, imaginative scientist, and we look forward to a continuing body of excellent work. Meanwhile, we wish him many more years of a happy and healthy life.

¹Wu met Tony Guttmann at the University of Newcastle, Australia, back in 1973 when Guttmann invited him to visit. Over the years their paths have crossed countless times at conferences and workshops, and during Wu's visits to Australia and Guttmann's to America; their families became close friends in the process, with Wu's wife Jane assisting Guttmann's wife Susette in her professional duties when they both visited Taiwan.

²Wu met Larry Glasser in 1968 at MIT and also visited him later at Clarkson. They collaborated in 2003 on a paper later published in the Ramanujan Journal in 2005, in which they evaluated an integral for the entropy of spanning trees on the triangular lattice.

³Wu and Jean-Marie Maillard enjoyed joint research grants, organised between the NSF and the CNRS. They also got together frequently in Taiwan and at conferences including one in Paris on the Yang–Baxter equation in 1992. They have many joint papers, including one of Wu's favorites, a 1992 J. Phys. A: Math. Gen. paper on thermal transmissivity. In that paper they put the loosely defined term transmissivity onto a rigorous footing.

⁴Henk Blöte and Wu first met in 1973 in Delft. Since then they have visited each other frequently, as Blöte made regular visits to the University of Rhode Island (near Boston) and Beijing Normal University, intersecting those of Wu. They first collaborated in a 1989 paper in which they obtained a closed-form expression for the critical curve of the honeycomb antiferromagnetic Ising model and checked the formula against finite-size analysis. This combination of checking an a priori derivation against high-precision numerical analysis set the tone of Wu's later collaborations with Blöte and his students.

⁵Youjin Deng obtained his PhD in 2004 under the direction of Blöte at Delft. Wu served on Deng's Dissertation Committee and participated in his graduation ceremony.

⁶Through Wu's recent works on the Potts model he got to know Bob Ziff well. They exchanged preprints and e-mails, and often had lengthy discussions on minute points, including the use and origin of the term 'hemp-leaf lattice'.

⁷Wu and Murray Batchelor met at the Australia National University in 1990 and again in 1995, and their paths have crossed at many conferences and workshops.

⁸Wenan Guo likewise obtained his PhD under the supervision of Blöte in Delft. Wu and Guo know each other well from Wu's visits to the Beijing Normal University where he is an honorary professor. He has collaborated with Guo, on the subject of finite-size analysis using the transfer matrix approach, in several of his recent papers.

⁹Wu first met John Essam at King's College, London in 1978. Followoing Wu's 2006 closed-form expression of the corner-to-corner resistance of an M × N resistor network in the form of a double summation, they combined forces in 2008 at a workshop in Cambridge, and derived the asymptotic expansion of that expression.

¹⁰Nickolay Izmailian holds positions in Armenia and Taiwan. Wu and Izmailian collaborated in a paper in 2000 on the exact solution of a 6-vertex model with bond defects. Most recently they collaborated on the exact enumeration of dimers on a cylinder with a single boundary defect.

¹¹Wu's acquaintance with Jesper Jacobsen goes back to this period, when the latter pointed out this fact in a comment to Wu's first paper on this subject. They have since crossed paths on various occasions, most recently at a 2008 workshop at the Isaac Newton Institute in Cambridge.

¹²Jerry Barry is another long-term collaborator of Wu's. They have met at numerous conferences and workshops. In one meeting in 1989, Barry called Wu's attention to a three-dimensional spin model on the pyrochlore lattice that appeared to be soluble. They soon solved the Ising model on that lattice. In 1997 they collaborated on a paper obtaining the phase diagram of a ternary polymer model.

We define the notion of a spanning tree generating function (STGF) ∑a_nzⁿ, which gives the spanning tree constant when evaluated at z = 1, and gives the lattice Green function (LGF) when differentiated. By making use of known results for logarithmic Mahler measures of certain Laurent polynomials, and proving new results, we express the STGFs as hypergeometric functions for all regular two and three dimensional lattices (and one higher-dimensional lattice). This gives closed form expressions for the spanning tree constants for all such lattices, which were previously largely unknown in all but one three-dimensional case. We show for all lattices that these can also be represented as Dirichlet L-series. Making the connection between STGFs and LGFs produces integral identities and hypergeometric connections, some of which appear to be new.

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 Mahler measure for the n-variable polynomial k + ∑(x_j + 1/x_j) is reduced to a single integral of the n-th power of the modified Bessel function I₀. Several special cases are examined in detail.

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.

Any two-dimensional infinite regular lattice G can be produced by tiling the plane with a finite subgraph B⊆G; we call B a basis of G. We introduce a two-parameter graph polynomial P_B(q, v) that depends on B and its embedding in G. The algebraic curve P_B(q, v) = 0 is shown to provide an approximation to the critical manifold of the q-state Potts model, with coupling v = e^K − 1, defined on G. This curve predicts the phase diagram not only in the physical ferromagnetic regime (v > 0), but also in the antiferromagnetic (v < 0) region, where analytical results are often difficult to obtain. For larger bases B the approximations become increasingly accurate, and we conjecture that P_B(q, v) = 0 provides the exact critical manifold in the limit of infinite B. Furthermore, for some lattices G—or for the Ising model (q = 2) on any G—the polynomial P_B(q, v) factorizes for any choice of B: the zero set of the recurrent factor then provides the exact critical manifold. In this sense, the computation of P_B(q, v) can be used to detect exact solvability of the Potts model on G. We illustrate the method for two choices of G: the square lattice, where the Potts model has been exactly solved, and the kagome lattice, where it has not. For the square lattice we correctly reproduce the known phase diagram, including the antiferromagnetic transition and the singularities in the Berker–Kadanoff phase at certain Beraha numbers. For the kagome lattice, taking the smallest basis with six edges we recover a well-known (but now refuted) conjecture of F Y Wu. Larger bases provide successive improvements on this formula, giving a natural extension of Wu's approach. We perform large-scale numerical computations for comparison and find excellent agreement with the polynomial predictions. For v > 0 the accuracy of the predicted critical coupling v_c is of the order 10⁻⁴ or 10⁻⁵ for the six-edge basis, and improves to 10⁻⁶ or 10⁻⁷ for the largest basis studied (with 36 edges).

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.

Percolation thresholds have recently been studied by means of a graph polynomial P_B(p), henceforth referred to as the critical polynomial, that may be defined on any periodic lattice. The polynomial depends on a finite subgraph B, called the basis, and the way in which the basis is tiled to form the lattice. The unique root of P_B(p) in [0, 1] either gives the exact percolation threshold for the lattice, or provides an approximation that becomes more accurate with appropriately increasing size of B. Initially P_B(p) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give an alternative probabilistic definition of P_B(p), which allows for much more efficient computations, by using the transfer matrix, than was previously possible with contraction-deletion. We present bond percolation polynomials for the (4, 8²), kagome, and (3, 12²) lattices for bases of up to respectively 96, 162 and 243 edges, much larger than the previous limit of 36 edges using contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. For the largest bases, we obtain the thresholds p_c(4, 8²) = 0.676 803 329..., p_c(kagome) = 0.524 404 998..., p_c(3, 12²) = 0.740 420 798..., comparable to the best simulation results. We also show that the alternative definition of P_B(p) can be applied to study site percolation problems.

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.

We give a conditional derivation of the inhomogeneous critical percolation manifold of the bow-tie lattice with five different probabilities, a problem that does not appear at first to fall into any known solvable class. Although our argument is mathematically rigorous only on a region of the manifold, we conjecture that the formula is correct over its entire domain, and we provide a non-rigorous argument for this that employs the negative probability regime of the triangular lattice critical surface. We discuss how the rigorous portion of our result substantially broadens the range of lattices in the solvable class to include certain inhomogeneous and asymmetric bow-tie lattices, and that, if it could be put on a firm foundation, the negative probability portion of our method would extend this class to many further systems, including F Y Wu's checkerboard formula for the square lattice. We conclude by showing that this latter problem can in fact be proved using a recent result of Grimmett and Manolescu for isoradial graphs, lending strong evidence in favor of our other conjectured results.

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.

We study the bond percolation problem under the constraint that the total number of occupied bonds is fixed, so that the canonical ensemble applies. We show via an analytical approach that at criticality, the constraint can induce new finite-size corrections with exponent y_can = 2y_t − d both in energy-like and magnetic quantities, where y_t = 1/ν is the thermal renormalization exponent and d is the spatial dimension. Furthermore, we find that while most of the universal parameters remain unchanged, some universal amplitudes, like the excess cluster number, can be modified and become non-universal. We confirm these predictions by extensive Monte Carlo simulations of the two-dimensional percolation problem which has y_can = −1/2.

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.

We study the combinatorics of the change of basis of three representations of the stationary state algebra of the two parameter simple asymmetric exclusion process. Each of the representations considered correspond to a different set of weighted lattice paths which, when summed over, give the stationary state probability distribution. We show that all three sets of paths are combinatorially related via sequences of bijections and sign reversing involutions.

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 generating function for the number of non-crossing walk configurations of n walks between the roots of a two-rooted directed plane graph is introduced. This is shown to be a rational function and the structure and symmetry property of its numerator are discussed. The walk configurations correspond to flows and the equivalent dual generating function for potentials is investigated independently. Also equivalences are drawn with the partially order sets that can be constructed from the walk configurations. Finally, the general results developed here are applied to the directed square and honeycomb lattices.

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.

We analyze the exact partition function of the anisotropic Ising model on finite M × N rectangular lattices under four different boundary conditions (periodic–periodic (pp), periodic–antiperiodic (pa), antiperiodic–periodic (ap) and antiperiodic–antiperiodic (aa)) obtained by Kaufman (1949 Phys. Rev.76 1232), Wu and Hu (2002 J. Phys. A: Math. Gen.35 5189) and Kastening (2002 Phys. Rev. E 66 057103)). We express the partition functions in terms of the partition functions Z_{α, β}(J, k) with (α, β) = (0, 0), (1/2, 0), (0, 1/2) and (1/2, 1/2), J is an interaction coupling and k is an anisotropy parameter. Based on such expressions, we then extend the algorithm of Ivashkevich et al (2002 J. Phys. A: Math. Gen.35 5543) to derive the exact asymptotic expansion of the logarithm of the partition function for all boundary conditions mentioned above. Our result is f = f_bulk + ∑^∞_{p = 0}f_p(ρ, k)S^{−p − 1}, where f is the free energy of the system, f_bulk is the free energy of the bulk, S = MN is the area of the lattice and ρ = M/N is the aspect ratio. All coefficients in this expansion are expressed through analytical functions. We have introduced the effective aspect ratio ρ_eff = ρ/sinh 2J_c and show that for pp and aa boundary conditions all finite size correction terms are invariant under the transformation ρ_eff → 1/ρ_eff.

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.

Focusing on examples associated with holonomic functions, we try to bring new ideas on how to look at phase transitions, for which the critical manifolds are not points but curves depending on a spectral variable, or even fill higher dimensional submanifolds. Lattice statistical mechanics often provides a natural (holonomic) framework to perform singularity analysis with several complex variables that would, in the most general mathematical framework, be too complex, or simply could not be defined. In a learn-by-example approach, considering several Picard–Fuchs systems of two-variables 'above' Calabi–Yau ODEs, associated with double hypergeometric series, we show that D-finite (holonomic) functions are actually a good framework for finding properly the singular manifolds. The singular manifolds are found to be genus-zero curves. We then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics, the n-fold integrals χ⁽ⁿ⁾, corresponding to the n-particle decomposition of the magnetic susceptibility of the anisotropic square Ising model. In this anisotropic case, we revisit a set of so-called Nickelian singularities that turns out to be a two-parameter family of elliptic curves. We then find the first set of non-Nickelian singularities for χ⁽³⁾ and χ⁽⁴⁾, that also turns out to be rational or elliptic curves. We underline the fact that these singular curves depend on the anisotropy of the Ising model, or, equivalently, that they depend on the spectral parameter of the model. This has important consequences on the physical nature of the anisotropic χ⁽ⁿ⁾s which appear to be highly composite objects. We address, from a birational viewpoint, the emergence of families of elliptic curves, and that of Calabi–Yau manifolds on such problems. We also address the question of singularities of non-holonomic functions with a discussion on the accumulation of these singular curves for the non-holonomic anisotropic full susceptibility χ.

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.

We consider a triangular gap of side 2 in a 60° angle on the triangular lattice whose sides are zigzag lines. We study the interaction of the gap with the corner as the rest of the angle being completely filled with lozenges. We show that the resulting correlation is governed by the product of the distances between the gap and its five images in the sides of the angle. This provides a new aspect of the parallel between the correlation of gaps in dimer packings and electrostatics developed by the first author in previous work.

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 problem of enumerating close-packed dimers, or perfect matchings, on two types of lattices (the so-called 8.8.4 and 8.8.6 lattices) embedded on the Klein bottle is considered, and we obtain the explicit expression of the number of close-packed dimers and entropy. Our results imply that 8.8.4 lattices have the same entropy under three different boundary conditions (cylindrical, toroidal and Klein bottle) and 8.8.6 lattices have the same property.

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.

Parafermionic observables were introduced by Smirnov for planar Fortuin–Kasteleyn (FK) percolation in order to study the critical phase (p, q) = (p_c(q), q). This paper gathers several known properties of these observables. Some of these properties are used to prove the divergence of the correlation length when approaching the critical point for FK percolation when 1 ⩽ q ⩽ 4. A crucial step is to consider FK percolation on the universal cover of the punctured plane. We also mention several conjectures on FK percolation with arbitrary cluster weight q > 0.

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.

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 Z_N 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.

We prove the existence of the Green function for radial SLE_κ for κ < 8. Unlike the chordal case where an explicit formula for the Green function is known for all values of κ < 8, we give an explicit formula only for κ = 4. For other values of κ, we give a formula in terms of an expectation with respect to SLE conditioned to go through a point.

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.

We present an investigation of the completely packed O(n) loop model on the square lattice by means of the transfer-matrix method and finite-size scaling. We investigate the model for a number of n values covering a wide range. This model is known to be equivalent with the q-state Potts model with q = n², but here we also investigate the range n < 0, including rather large negative numbers. In the critical range |n| < 2, we find an energy-like scaling dimension X = 4, which is the leading one for n < 1 and the second leading one for 1 < n < 2. The point n = −2 is special, with a conformal anomaly c = −∞. For n < −2, the model is no longer critical, as evidenced e.g. by the exponentially fast convergence of the finite-size estimates of the free energy density to the infinite-system value. For |n| > 2, the system is in an ordered phase, where the majority of the loops cover part of the elementary faces of the lattice in one of two checkerboard patterns that are in phase coexistence. Furthermore, we find that the numerical results for the free energy density are in agreement with the expressions obtained from the exact analysis of the equivalent six-vertex model.

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.

We use the nested loop approach to investigate loop models on random planar maps where the domains delimited by the loops are given two alternating colors, which can be assigned different local weights, hence allowing for an explicit Z₂ domain symmetry breaking. Each loop receives a non-local weight n, as well as a local bending energy which controls loop turns. By a standard cluster construction that we review, the Q = n² Potts model on general random maps is mapped to a particular instance of this problem with domain-non-symmetric weights. We derive in full generality a set of coupled functional relations for a pair of generating series which encode the enumeration of loop configurations on maps with a boundary of a given color, and solve it by extending well-known complex analytic techniques. In the case where loops are fully packed, we analyze in detail the phase diagram of the model and derive exact equations for the position of its non-generic critical points. In particular, we underline that the critical Potts model on general random maps is not self-dual whenever Q ≠ 1. In a model with domain-symmetric weights, we also show the possibility of a spontaneous domain symmetry breaking driven by the bending energy.

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.

We give the derivation of the previously announced analytic expression for the correlation function of three heavy non-BPS operators in ${ \mathcal N}=4$ super-Yang–Mills theory at weak coupling. The three operators belong to three different su(2) sectors and are dual to three classical strings moving on the sphere. Our computation is based on the reformulation of the problem in terms of the Bethe ansatz for periodic XXX spin-1/2 chains. In these terms, the three operators are described by long-wavelength excitations over the ferromagnetic vacuum, for which the number of the overturned spins is a finite fraction of the length of the chain, and the classical limit is known as the Sutherland limit. Technically, our main result is a factorized operator expression for the scalar product of two Bethe states. The derivation is based on a fermionic representation of Slavnov's determinant formula, and a subsequent bosonization.

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.

A system of mesoscopic ions with dominant three-particle interactions is modeled by a quantum lattice liquid on the planar kagomé lattice. The two-parameter Hamiltonian contains localized attractive triplet interactions as potential energy and nearest neighbor hopping-type terms as kinetic energy. The dynamic ionic conductivity σ(ω) is theoretically investigated for 'weak hopping' via a quantum many-body perturbation expansion of the thermal (Matsubara) Green function (current-current correlation). A simple analytic continuation and mapping of the thermal Green function provide the temporal Fourier transform of the physical retarded Green function in the Kubo formula. Substituting pertinent exact solutions for static multi-particle correlations known from previous work, Arrhenius relations are revealed in zeroth-order approximation for the dc ionic conductivity σ_dc along special trajectories in density–temperature space. The Arrhenius plots directly yield static activation energies along the latter loci. Experimental possibilities relating to σ_dc are discussed in the presence of equilibrium aggregation.

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.

We describe a measurement device principle based on discrete iterations of Bayesian updating of system-state probability distributions. Although purely classical by nature, these measurements are accompanied with a progressive collapse of the system-state probability distribution during each complete system measurement. This measurement scheme finds applications in analysing repeated non-demolition indirect quantum measurements. We also analyse the continuous time limit of these processes, either in the Brownian diffusive limit or in the Poissonian jumpy limit. In the quantum mechanical framework, this continuous time limit leads to Belavkin's equations which describe quantum systems under continuous measurements.

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.