Hitting Matrix and Domino Tiling with Diagonal Impurities

As a continuation to our previous work (Nakano and Sadahiro in Fundam. Inform. 117:249–264, 2012; Nakano and Sadahiro in J. Stat. Phys. 139(4):565–597, 2010), we consider the domino tiling problem with impurities. (1) If we have more than two impurities on the boundary, we can compute the number of corresponding perfect matchings by using the hitting matrix method (Fomin in Trans. Am. Math. Soc. 353(9):3563–3583, 2001). (2) We have an alternative proof of the main result in Nakano and Sadahiro (Fundam. Inform. 117:249–264, 2012) and result in (1) above using the formula by Kenyon and Wilson (Trans. Am. Math. Soc. 363(3):1325–1364, 2011; Electron. J. Comb. 16(1):112, 2009) of counting the number of groves on circular planar graphs. (3) We study the behavior of the probability of finding the impurity at a given site when the size of the graph tends to infinity, as well as the scaling limit of those.     

Quantum Random Walks with General Particle States

A convergence theorem is obtained for quantum random walks with particles in an arbitrary normal state. This unifies and extends previous work on repeated-interactions models, including that of Attal and Pautrat (Ann Henri Poincaré 7:59–104 2006) and Belton (J Lond Math Soc 81:412–434, 2010; Commun Math Phys 300:317–329, 2010). When the random-walk generator acts by ampliation and either multiplication or conjugation by a unitary operator, it is shown that the quantum stochastic cocycle which arises in the limit is driven by a unitary process.     

Asymptotic Expansion of β Matrix Models in the One-cut Regime

We prove the existence of a 1/N expansion to all orders in β matrix models with a confining, offcritical potential corresponding to an equilibrium measure with a connected support. Thus, the coefficients of the expansion can be obtained recursively by the “topological recursion” derived in Chekhov and Eynard (JHEP 0612:026, 2006). Our method relies on the combination of a priori bounds on the correlators and the study of Schwinger-Dyson equations, thanks to the uses of classical complex analysis techniques. These a priori bounds can be derived following (Boutet de Monvel et al. in J Stat Phys 79(3–4):585–611, 1995; Johansson in Duke Math J 91(1):151–204, 1998; Kriecherbauer and Shcherbina in Fluctuations of eigenvalues of matrix models and their applications, 2010) or for strictly convex potentials by using concentration of measure (Anderson et al. in An introduction to random matrices, Sect. 2.3, Cambridge University Press, Cambridge, 2010). Doing so, we extend the strategy of Guionnet and Maurel-Segala (Ann Probab 35:2160–2212, 2007), from the hermitian models (β = 2) and perturbative potentials, to general β models. The existence of the first correction in 1/N was considered in Johansson (1998) and more recently in Kriecherbauer and Shcherbina (2010). Here, by taking similar hypotheses, we extend the result to all orders in 1/N.     

A Note on Permutation Twist Defects in Topological Bilayer Phases

We present a mathematical derivation of some of the most important physical quantities arising in topological bilayer systems with permutation twist defects as introduced by Barkeshli et al. (Phys Rev B 87:045130_1-23, 2013). A crucial tool is the theory of permutation equivariant modular functors developed by Barmeier et al. (Int Math Res Notices 2010:3067–3100, 2010; Transform Groups 16:287–337, 2011).     

The Parafermionic Observable in SLE

The parafermionic observable has recently been used by number of authors to study discrete models, believed to be conformally invariant and to prove convergence results for these processes to SLE (Beffara and Duminil-Copin in arXiv:1010.0526v2, 2011; Duminil-Copin and Smirnov in arXiv:1007.0575v2, 2011; Hongler and Smirnov in arXiv:1008.2645v3, 2011; Ikhlef and Cardy in J. Phys. A 42:102001, 2009; Lawler in preprint, 2011; Rajabpour and Cardy in J. Phys. A 40:14703, 2007; Riva and Cardy in J. Stat. Mech. Theory Exp., 2006; Smirnov in International Congress of Mathematicians, vol. II, pp. 1421?C1451, 2006; Smirnov in Ann. Math. 172(2):1435?C1467, 2010; Smirnov in Proceedings of the International Congress of Mathematicians, Hyderabad 2010, vol.?I, pp. 595?C621, 2010). We provide a definition for a one parameter family of continuum versions of the parafermionic observable for SLE, which takes the form of a normalized limit of expressions identical to the discrete definition. We then show the limit defining the observable exists, compute the value of the observable up to a finite multiplicative constant, and prove this constant is non-zero for a wide range of ??. Finally, we show our observable for SLE becomes a holomorphic function for a particular choice of the parameter, which provides a new point of view on a fundamental property of the discrete observable.     

Note on Solutions of the Strominger System from Unitary Representations of Cocompact Lattices of {SL(2,\mathbb{C})}

This note is motivated by a recently published paper (Biswas and Mukherjee in Commun Math Phys 322(2):373–384, 2013). We prove a no-go result for the existence of suitable solutions of the Strominger system in a compact complex parallelizable manifold \({M = G/\Gamma}\) . For this, we assume G to be non-abelian, the Hermitian metric to be induced from a right invariant metric on G, the Bianchi identity to be satisfied using the Chern connection and furthermore the gauge field to be flat. In Biswas and Mukherjee (Commun Math Phys 322(2):373–384, 2013) it is claimed that one such solution exists on \({SL(2, \mathbb{C})/\Gamma}\) . Our result contradicts the main result in Biswas and Mukherjee (Commun Math Phys 322(2):373–384, 2013).     

Tracy–Widom at High Temperature

We investigate the marginal distribution of the bottom eigenvalues of the stochastic Airy operator when the inverse temperature \(\beta \) tends to \(0\) . We prove that the minimal eigenvalue, whose fluctuations are governed by the Tracy–Widom \(\beta \) law, converges weakly, when properly centered and scaled, to the Gumbel distribution. More generally we obtain the convergence in law of the marginal distribution of any eigenvalue with given index \(k\) . Those convergences are obtained after a careful analysis of the explosion times process of the Riccati diffusion associated to the stochastic Airy operator. We show that the empirical measure of the explosion times converges weakly to a Poisson point process using estimates proved in Dumaz and Virág (Ann Inst H Poincaré Probab Statist 49(4):915–933, 2013). We further compute the empirical eigenvalue density of the stochastic Airy ensemble on the macroscopic scale when \(\beta \rightarrow 0\) . As an application, we investigate the maximal eigenvalues statistics of \(\beta _N\) -ensembles when the repulsion parameter \(\beta _N\rightarrow 0\) when \(N\rightarrow +\infty \) . We study the double scaling limit \(N\rightarrow +\infty , \beta _N \rightarrow 0\) and argue with heuristic and numerical arguments that the statistics of the marginal distributions can be deduced following the ideas of Edelman and Sutton (J Stat Phys 127(6):1121–1165, 2007) and Ramírez et al. (J Am Math Soc 24:919–944, 2011) from our later study of the stochastic Airy operator.     

Addendum to: A Renormalizable 4-Dimensional Tensor Field Theory

This note fills a gap in the article (Ben Geloun and Rivasseau, Commun Math Phys 318:69–109, 2013). We provide the proof of Eq. (82) of Lemma 5 in Ben Geloun and Rivasseau (Commun Math Phys 318:69–109, 2013) and thereby complete its power-counting analysis with a more precise next-to-leading-order estimate.     

A Few Endpoint Geodesic Restriction Estimates for Eigenfunctions

We prove a couple of new endpoint geodesic restriction estimates for eigenfunctions. In the case of general 3-dimensional compact manifolds, after a TT* argument, simply by using the L ²-boundedness of the Hilbert transform on ${\mathbb{R}}$ , we are able to improve the corresponding L ²-restriction bounds of Burq, Gérard and Tzvetkov (Duke Math J 138:445–486, 2007) and Hu (Forum Math 6:1021–1052, 2009). Also, in the case of 2-dimensional compact manifolds with nonpositive curvature, we obtain improved L ⁴-estimates for restrictions to geodesics, which, by Hölder’s inequality and interpolation, implies improved L ^p -bounds for all exponents p ≥ 2. We do this by using oscillatory integral theorems of Hörmander (Ark Mat 11:1–11, 1973), Greenleaf and Seeger (J Reine Angew Math 455:35–56, 1994) and Phong and Stein (Int Math Res Notices 4:49–60, 1991), along with a simple geometric lemma (Lemma 3.2) about properties of the mixed-Hessian of the Riemannian distance function restricted to pairs of geodesics in Riemannian surfaces. We are also able to get further improvements beyond our new results in three dimensions under the assumption of constant nonpositive curvature by exploiting the fact that, in this case, there are many totally geodesic submanifolds.     

Quantizing the Discrete Painlevé VI Equation: The Lax Formalism

A discretization of Painlevé VI equation was obtained by Jimbo and Sakai (Lett Math Phys 38:145–154, 1996). There are two ways to quantize it: (1) use the affine Weyl group symmetry (of ${D_5^{(1)}}$ ) (Hasegawa in Adv Stud Pure Math 61:275–288, 2011), (2) Lax formalism, i.e. monodromy preserving point of view. It turns out that the second approach is also successful and gives the same quantization as in the first approach.     

n-Orthodistributivity in Orthomodular Lattices

A useful generalization of distributivity in lattices n-distributivity, \(n \in \mathbb{N}\) , was introduced in Huhn (Acta Sci. Math. 33:297–305, 1972). In Mayet and Roddy (Contrib. Gen. Algebra 5:285–294, 1987), ‘orthogonalized’ versions, n-orthodistributivity, \(n \in \mathbb{N}\) , of these equations were introduced and discussed. The discussion and results of Mayet and Roddy (Contrib. Gen. Algebra 5:285–294, 1987) centered on the class of modular ortholattices. In this paper we discuss and present some preliminary results for these conditions in orthomodular lattices. In particular, we completely classify the n-(ortho)distributive orthomodular lattices arising from Greechie’s classical 1971 construction, and we prove that a certain simple atomless orthomodular lattice, presented in Roddy (Algebra Univers. 29:564–597, 1992), is 4-orthodistributive. It is not 3-orthodistributive.     

Perturbed GUE Minor Process and Warren’s Process with Drifts

We consider the minor process of (Hermitian) matrix diffusions with constant diagonal drifts. At any given time, this process is determinantal and we provide an explicit expression for its correlation kernel. This is a measure on the Gelfand–Tsetlin pattern that also appears in a generalization of Warren’s process (Electron. J. Probab. 12:573–590, 2007), in which Brownian motions have level-dependent drifts. Finally, we show that this process arises in a diffusion scaling limit from an interacting particle system in the anisotropic KPZ class in 2+1 dimensions introduced in Borodin and Ferrari (Commun. Math. Phys., 2008). Our results generalize the known results for the zero drift situation.     

From Fixed-Energy Localization Analysis to Dynamical Localization: An Elementary Path

We review several techniques initiated by a remarkable work written by Spencer (J Stat Phys 51:1009–1019, 1988) a quarter of century ago, used and further developed in numerous subsequent researches. We also describe a fairly general, elementary derivation of spectral and strong dynamical Anderson localization from the fixed-energy analysis of the Green’s functions on locally finite graphs of polynomial growth, obtained either by the multi-scale analysis or by the fractional-moment method. This derivation goes in the same direction as the Simon–Wolf method (Commun Pure Appl Math 39:75–90, 1986), but provides more quantitative estimates, can be adapted to multi-particle models and, combined with a simplified variant of the Germinet–Klein argument (Commun Math Phys 222:415–448, 2001), results in an elementary proof of strong dynamical localization on arbitrary graphs of polynomial growth.     

Bifurcation analysis and the travelling wave solutions of the Klein–Gordon–Zakharov equations

In this paper, we investigate the bifurcations and dynamic behaviour of travelling wave solutions of the Klein–Gordon–Zakharov equations given in Shang et al, Comput. Math. Appl. 56, 1441 (2008). Under different parameter conditions, we obtain some exact explicit parametric representations of travelling wave solutions by using the bifurcation method (Feng et al, Appl. Math. Comput. 189, 271 (2007); Li et al, Appl. Math. Comput. 175, 61 (2006)).     

From Cycle Rooted Spanning Forests to the Critical Ising Model: an Explicit Construction

Fisher established an explicit correspondence between the 2-dimensional Ising model defined on a graph G and the dimer model defined on a decorated version ${\mathcal{G}}$ of this graph (Fisher in J Math Phys 7:1776–1781, 1966). In this paper we explicitly relate the dimer model associated to the critical Ising model and critical cycle rooted spanning forests (CRSFs). This relation is established through characteristic polynomials, whose definition only depends on the respective fundamental domains, and which encode the combinatorics of the model. We first show a matrix-tree type theorem establishing that the dimer characteristic polynomial counts CRSFs of the decorated fundamental domain ${\mathcal{G}_1}$ . Our main result consists in explicitly constructing CRSFs of ${\mathcal{G}_1}$ counted by the dimer characteristic polynomial, from CRSFs of G ₁, where edges are assigned Kenyon’s critical weight function (Kenyon in Invent Math 150(2):409–439, 2002); thus proving a relation on the level of configurations between two well known 2-dimensional critical models.     

Non-Equilibrium Statistical Mechanics of Turbulence

The macroscopic study of hydrodynamic turbulence is equivalent, at an abstract level, to the microscopic study of a heat flow for a suitable mechanical system (Ruelle, PNAS 109:20344–20346, 2012). Turbulent fluctuations (intermittency) then correspond to thermal fluctuations, and this allows to estimate the exponents \(\tau _p\) and \(\zeta _p\) associated with moments of dissipation fluctuations and velocity fluctuations. This approach, initiated in an earlier note (Ruelle, 2012), is pursued here more carefully. In particular we derive probability distributions at finite Reynolds number for the dissipation and velocity fluctuations, and the latter permit an interpretation of numerical experiments (Schumacher, Preprint, 2014). Specifically, if \(p(z)dz\) is the probability distribution of the radial velocity gradient we can explain why, when the Reynolds number \(\mathcal{R}\) increases, \(\ln p(z)\) passes from a concave to a linear then to a convex profile for large \(z\) as observed in (Schumacher, 2014). We show that the central limit theorem applies to the dissipation and velocity distribution functions, so that a logical relation with the lognormal theory of Kolmogorov (J. Fluid Mech. 13:82–85, 1962) and Obukhov is established. We find however that the lognormal behavior of the distribution functions fails at large value of the argument, so that a lognormal theory cannot correctly predict the exponents \(\tau _p\) and \(\zeta _p\) .     

Tail Asymptotics of Free Path Lengths for the Periodic Lorentz Process: On Dettmann’s Geometric Conjectures

In the simplest case, consider a \({\mathbb{Z}^d}\) -periodic (d ≥ 3) arrangement of balls of radii < 1/2, and select a random direction and point (outside the balls). According to Dettmann’s first conjecture, the probability that the so determined free flight (until the first hitting of a ball) is larger than t >  > 1 is \({\sim\frac{C}{t}}\) , where C is explicitly given by the geometry of the model. In its simplest form, Dettmann’s second conjecture is related to the previous case with tangent balls (of radii 1/2). The conjectures are established in a more general setup: for \({\mathcal{L}}\) -periodic configuration of—possibly intersecting—convex bodies with \({\mathcal{L}}\) being a non-degenerate lattice. These questions are related to Pólya’s visibility problem (Arch Math Phys Ser 2:135–142, 1918), to theories of Bourgain et al. (Commun Math Phys 190:491–508,1998), and of Marklof–Strömbergsson (Ann Math 172:1949–2033,2010). The results also provide the asymptotic covariance of the periodic Lorentz process assuming it has a limit in the super-diffusive scaling, a fact if d = 2 and the horizon is infinite.     

Threshold Phenomenon for the Quintic Wave Equation in Three Dimensions

For the critical focusing wave equation ${\square u = u^5 \, {\rm on} \, \mathbb{R}^{3+1}}$ in the radial case, we establish the role of the “center stable” manifold ${\Sigma}$ constructed in Krieger and Schlag (Am J Math 129(3):843–913, 2007) near the ground state (W, 0) as a threshold between blowup and scattering to zero, establishing a conjecture going back to numerical work by Bizoń et al. (Nonlinearity 17(6):2187–2201, 2004). The underlying topology is stronger than the energy norm.     

Multilocal Fermionization

We present a simple isomorphism between the algebra of one real chiral Fermi field and the algebra of n real chiral Fermi fields. The isomorphism preserves the vacuum state. This is possible by a “change of localization”, and gives rise to new multilocal symmetries generated by the corresponding multilocal current and stress–energy tensor. The result gives a common underlying explanation of several remarkable recent results on the representation of the free Bose field in terms of free Fermi fields (Anguelova, arXiv:1112.3913, 2011; Anguelova, arXiv:1206.4026, 2012), and on the modular theory of the free Fermi algebra in disjoint intervals (Casini and Huerta, Class Quant Grav 26:185005, 2009; Longo et al., Rev Math Phys 22:331–354, 2010)