In parallel, **the** hermitian two-**matrix** **model** (2MM) solutions have been obtained, at early stages, in **the** planar limit of **the** 1/N-**expansion** [34,21]. It was however almost immediately observed [5, 18, 21] that **the** 2MM solution in **the** planar limit enjoys **the** same geometrical properties as **the** 1MM solution; only **the** spectral curve becomes an arbitrary algebraic curve, not just an hyperellitic curve arising in **the** 1MM case. **The** subsequent progress was however hindered by that **the** corresponding master loop equation in **the** 2MM case cannot immediately be expressed in terms of correlation functions alone. Nevertheless, using **the** geometrical properties of this equation, **the** solution in **the** first subleading order has been constructed [19, 20] on **the** base of knowledge of Bergmann tau function on Hurwitz spaces. However, **the** general belief arose that **the** 2MM case should not be very much different from **the** 1MM case, that is, we expect to find all ingredients of **the** 1MM solution in **the** 2MM case. Next step towards constructing **the** **topological** **expansion** of **the** 2MM was performed in [24], where **the** first variant of **the** diagrammatic technique **for** **the** correlation functions in **the** 2MM case was constructed. In **the** present paper we improve **the** technique of [24] (actually effectively simplifying it) to accommodate **the** action of **the** loop insertion operator. In fact, we demonstrate that **the** diagrammatic technique **for** **the** 2MM case is even closer to **the** one in **the** 1MM case that was before: in particular, we need only three-valent vertices, and **the** additional operator H we need to obtain **the** **free** **energy** turns out to be of **the** same origin as **the** one in **the** 1MM.

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of **the** matrices, and could not be computed by standard methods. **The** first explicit computations were obtained in [4] and [17]. Here in this paper, we show how to compute **the** **topological** **expansion** of a family of mixed correlation functions of **the** **2**-**matrix** **model**. In a coming work [23], we shall show how to compute all mixed correlations, and introduce a link with group theory and Bethe ansatz (this is a generalization of [22]).

Recently, it has progressively become clear that large N **expansion** of random **matrix** models has a strong link with algebraic geometry [27]. **The** **free** **energy** and correlation functions have been computed in terms of properties of an algebraic curve. **The** large N limit of **the** 1-point correlation function (called **the** resolvent) is solution of an algebraic equation, which thus defines an algebraic curve. There have been many works which computed **free** **energy** and correlation functions in terms of that algebraic curve. **The** leading order resolvent and **free** **energy** were computed in **the** 1-cut case (algebraic curve of genus zero) in **the** pioneering work of [7], then some recursive method **for** computing correlation functions and **free** **energy** to all orders in 1/N were invented by [3, 4]. Those methods were first limited to 1-**matrix** case and 1-cut.

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What is more interesting is to see what is Z N (E) **for** curves not coming from **the**
large N limit of **the** loop equations of a **matrix** **model**. We study in details a few examples.
- **The** double scaling limit of a **matrix** **model**. It has been well known since [51, 26], that if we fine tune **the** parameters of a **matrix** **model** so that **the** algebraic curve E develops a singularity, **the** **free** energies become singular and **the** most singular part of **the** **free** energies form **the** KP-hierarchy τ -function (KdV hierarchy **for** **the** 1-**matrix** **model**). We show, by looking at **the** double scaling limit of **matrix** models, that **the** KP τ -function (resp. KdV τ -function), coincides with our definition **for** **the** classical limit of **the** (p, q) systems (resp. (p, **2**)).

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Let us also remark that **the** function eq.**2**-27 E(x, y) is unchanged under **the** exchange x ↔ y, V1 ↔ V2, s ↔ ˜ s, which is **the** generalization of Matytsin’s duality property [17]:
X(Y (x)) = x (4-1)
**The** consequences of that algebraic equation, can then be studied. This is done **for** instance in [3]. One can also expect to generalize **the** works of [9], or [7], or [8], i.e. **the** computation of all correlation functions and their 1/N **2** **expansion**, and further, compute **the** **expansion** of **the** **free** **energy** [10, 11].

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fields also have some combinatorial interpretations, and have been studied **for** various applications. **The** most famous is **the** Kontsevich integral, which is **the** generating function **for** intersection numbers [33, 24, 19].
Here, we solve this more general **model**.
Multimatrix **model** also play an important role in quantum gravity and string theory, where they play **the** role of a regularized discrete space-time. **The** 1-**matrix** **model**, counts discrete surfaces without color, and is a **model** **for** quantum gravity without matter, whereas **the** chain of matrices counts discrete surfaces with n colors, and is interpreted as a **model** of quantum gravity with some matter field [27, 10, 1, 29, 30, 13], namely a matter which can have n possible states. More recently, **matrix** models have played a role in **topological** string theory [12].

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modeled as a rigid body and **the** tube as a deformable material. In **the** **model**, a surface-to-surface contact algorithm was ap- plied to **the** interface between **the** tube and **the** die with Coulomb’s friction set to 0.05 in accordance with different reported values in **the** literature **for** an unlubricated condition [ 4 , 22 ]. To mimic **the** experimental loading conditions in **the** FE **model**, **the** internal pressure was applied on each element of **the** meshed tube and increased linearly, while fixing **the** end nodes of **the** tube to emulate **the** no end feeding condition used during **the** **free** **expansion** process. **The** simulation results **for** **the** bulge height versus internal pressure were computed up to **the** maximum bulge height obtained experimentally. It is noteworthy that in **the** present work, **the** effect of anisotropy and, in **the** case of SS 304L only, **the** welded seam on **the** mechanical response was not considered in **the** FE **model** of **the** **free** **expansion** process. To this end, **the** effective true stress–true stain data employed in **the** FE **model** were based on **the** formulations given in Table **2** (spline profile) **for** **the** different alloys studied in this work. To understand **the** influ- ence of different material models on **the** FE simulation results, **the** uniaxial true stress–true strain curves as formulated in Table 3 were also implemented in **the** **free** **expansion** **model**. In addition **the** influence of **the** bulge profile on **the** FE simulation results was considered using **the** formulations in Table 4 (elliptical profile).

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These results highlight **the** benefits of ISCD-dependent Born radii to account **for** charge asymmetry effects, as well as **the** improvements afforded by training on real molecules **for** **the** nonlinear **model**. **The** linear correction function appears to provide robust and competitive results when compared with **the** nonlinear correction functions. However, as presented in **the** Theory and Implementation section, our computational experiments on spheres clearly show that correction should be nonlinear with respect to **the** induced surface charge density. **The** linear function most likely gives good results since **the** ISCD range explored by **the** neutral molecules in our data sets is rather narrow (between -0.01 and +0.01 e/Å **2** , see Figure S2, Supporting Information), **for** which **the** linear approximation is still applicable (see Figure **2**). With charged molecules, **the** range of ISCD will be expanded and **the** linear correction will most likely fail. **For** example, **the** nitrogen of a terminal alkyl ammonium would have an ISCD of around -0.025 e/Å **2** , which falls outside of **the** linear region seen in Figure **2** and justifies **the** use of a nonlinear function. A more complete study (outside **the** scope of this work) is needed **for** charged molecules, but **for** now, **the** nonlinear **model** seems most appropriate because of its greater generality. Given **the** comparable performances obtained with **the** two nonlinear correction functions, **the** rational function (eq 5) is preferred over **the** arctan + Gaussian function (eq 4) due to a lower number of fitted parameters and will be featured **for** **the** rest of **the** paper. **The** correlation between **the** FiSH continuum electrostatic component and **the** explicit-solvent LIE electrostatic term **for** **the** training, testing, and SAMPL1 data sets is shown in Figure 4.

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Math´ ematiques, Montr´ eal. **2**
? Institut de physique th´ eorique, CEA Saclay, France. 3
Abstract: **The** goal of this article is to rederive **the** connection between **the** Painlev´ e 5 integrable system and **the** universal eigenvalues correlation functions of double-scaled hermi- tian **matrix** models, through **the** **topological** recursion method. More specifically we prove, to all orders, that **the** WKB asymptotic expansions of **the** τ -function as well as of determinan- tal formulas arising from **the** Painlev´ e 5 Lax pair are identical to **the** large N double scaling asymptotic expansions of **the** partition function and correlation functions of any hermitian **matrix** **model** around a regular point in **the** bulk. In other words, we rederive **the** “sine-law” universal bulk asymptotic of large random matrices and provide an alternative perturbative proof of universality in **the** bulk with only algebraic methods. Eventually we exhibit **the** first orders of **the** series **expansion** up to O(N −5 ).

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4.1. Phase diagram
Figure 1 shows **the** **topological** phase diagram **for** different values of interaction U , combining results from **the** mean-field analysis and DMRG simulations. **The** agreement between both approaches is overall very good. **The** main standout feature of **the** phase diagram is that **the** **topological** phase and its associated Majorana bound states at zero **energy** are robust to interactions. In fact, **the** parameter range **for** which a **topological** Majorana phase is stabilized globally expands, upon increasing **the** Hubbard coupling. However, there is also a region in **the** phase diagram (|∆/w|, |g/w| < 1), where **the** Hubbard interaction is detrimental to **the** Majoranas. **The** mean-field phase diagram captures **the** **topological** transitions of **the** **model** rather accurately **for** small and moderate interactions. At strong interactions, **the** mean field keeps a very good estimate of **the** **topological** transition at small g/w, while at large g/w it tends to overestimate **the** extension of **the** Majorana phase. **The** “spin-orbit” coupling g tends to delocalize **the** electrons and leads to quantum fluctuations in **the** particle number. This explains, at a qualitative level, **the** divergence of **the** mean field results from **the** “exact” DMRG results at large “spin-orbit” coupling. A priori, this overall very good agreement may seem surprising in a quantum 1D system, where fluctuations are expected to be pronounced. However, let us recall that U(1) charge and **the** SU(**2**) spin-rotation symmetries are broken in this **model**, and hence International Conference on Strongly Correlated Electron Systems 2014 (SCES2014) IOP Publishing Journal of Physics: Conference Series 592 (2015) 012133 doi:10.1088/1742-6596/592/1/012133

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1.3 Outline of **the** paper and results
We will show that it is possible to build numerical schemes discretizing **the** Oldroyd-B system (1)– (**2**) such that solutions to those discretizations satisfy a **free** **energy** estimate similar to that estab- lished in [24, 20] **for** smooth solutions to **the** continuous equations. Our approach bears similarity with [36], where **the** authors also derive a discretization that preserves an **energy** estimate satisfied at **the** continuous level, and with [32], where another discretization is proposed **for** **the** same **energy** estimate as in [36]. Yet, unlike **the** estimates in [36, 32], our estimate, **the** so-called **free** **energy** estimate derived in [24, 20], ensures **the** long-time stability of solutions. As mentioned above, long-time computations are indeed often used to obtain a stationary state.

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CNRS, URA 2306, F-91191 Gif-sur-Yvette, France.
Abstract
We solve **the** loop equations of **the** β-ensemble **model** analogously to **the** solution found **for** **the** Hermitian matrices β = 1. **For** β = 1, **the** solution was expressed using **the** algebraic spectral curve of equation y **2** = U (x). **For** arbitrary β, **the** spectral curve converts into a Schr¨ odinger equation ((~∂) **2** −U(x))ψ(x) = 0 with ~ ∝ ( √ β −1/ √ β)/N . This paper is similar to **the** sister paper I, in particular, all **the** main ingredients specific **for** **the** algebraic solution of **the** problem remain **the** same, but here we present **the** second approach to finding a solution of loop equations using sectorwise definition of resolvents. Being technically more involved, it allows to define consistently **the** B-cycle structure of **the** obtained quantum algebraic curve (a D-module of **the** form y **2** −U(x), where [y, x] = ~) and to construct explicitly **the** correlation functions and **the** corresponding symplectic invariants F h , or **the** terms of **the** **free** **energy**, in 1/N **2** -**expansion** at arbitrary ~.

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Each Mk lies in **the** potential well Vk and is linearly coupled to its neigh- boors Mk−1 and Mk+1. M0 and MN have only one neighboor, and thus **the** chain is open. So far, **the** problem of **the** closed chain has remained unsolved. **The** multimatrix **model** is a generalization of **the** **2**-**matrix** **model**. It was often considered in **the** context of **2**-dimensional quantum gravity and string theory. Its critical points are known to represent **the** minimal conformal field theories, characterized by a pair of integers (p, q). It is known that one can get a (p, q) critical point with a multimatrix **model** where N = q − **2**. **The** necessity of studying multimatrix models can be understood from **the** fact that **the** one **matrix** models contains only **the** critical models with q ≤ **2**.

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A physical picture is that **the** support of **the** large N average density of eigenvalues of **the** **matrix** M 1 is made of g + 1 intervals [a i , b i ], i = 1, . . . , g + 1, and **for** each i, A i
is a contour which encloses [a i , b i ] in **the** trigonometric direction, and which does not
enclose **the** other a j or b j with j 6= i. eq. (3.3) means that **the** interval [a i , b i ] contains a

xTrpΦpxqq TrpΦpyqqy U β,n “ nGpx, yq.
4. Proofs
4.1. Proof of Theorem 3.2. We give **the** proof of Theorem 3.2, which contains Theorem 3.1 as a special case. A possible approach would be to develop **the** product of traces on **the** left-hand side of **the** isomorphism identity, apply Theorem 2.1, and then recombine **the** terms to get **the** right-hand side of **the** isomorphism identity. This works well **for** β P t1, 2u (see [Zvo97] **for** one **matrix** integrals), but **for** β “ 4 this approach is less tractable because of **the** non-commutativity of quaternions. Instead, we will rely on a recurrence over **the** numbers of edges |ν|{**2**, similar to that in [BP09].

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In this paper, we focus on such an effect of electron correlation on **the** phase structure of **topological** insulators, described by **the** boundary between **the** **topological** insulator and **the** normal insulator phases. We formulate **the** models of **topological** insulators in terms of massive lattice femions: in **2**-dimensions (2D), we take **the** Kane–Melé **model** **for** quantum spin Hall (QSH) in- sulator, one of **the** classes of **topological** insulators in 2D, which is a straightforward extension of **the** conventional tight-binding **model** of graphene. In 3D, we make use of **the** Wilson fermion on **the** hypothetical square lattice. As in **the** previous studies on graphene, we incorporate **the** electron-electron interaction mediated by **the** electromagnetic field in terms of lattice gauge theory, and analyze **the** emergent orders by **the** strong coupling **expansion** methods. As a result, we find a new phase with an in-plane antiferromagnetism in 2D, which appears by **the** similar mechanism as that of **the** pion condensate phase (so-called “Aoki phase”) in lattice QCD with Wilson fermion [9]. On **the** other hand, such a phase does not appear in 3D, and **the** electron correlation results in **the** shifting of **the** **topological** phase boundary [10].

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conjecture is that it provides a systematic method **for** computing **the** **topological** string partition function, genus by genus, away from **the** large radius limit, and without having to solve differential equations.
This conjecture was motivated by **the** fact that symplectic invariants have many intriguing properties reminiscent of **the** **topological** string **free** energies. They are invariant under transformations S → ˜ S which conserve **the** symplectic form dx ∧ dy = d˜ x ∧ d˜ y, whence their name [19]. They satisfy holomorphic anomaly equations [41], they have an integrable structure similar to Givental’s formulae [42, 43, 44, 45, 46], they satisfy some special geometry relations, WDVV relations [47], and they give **the** Witten-Kontsevich theory as a special case [19, 48].

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