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European Mathematical Society Publishing House
2024-03-29 15:05:48
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=AIHPD&vol=5&update_since=2024-03-29
Annales de l’Institut Henri Poincaré D
Ann. Inst. Henri Poincaré Comb. Phys. Interact.
AIHPD
2308-5827
2308-5835
General
Combinatorics
Quantum theory
10.4171/AIHPD
http://www.ems-ph.org/doi/10.4171/AIHPD
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
5
2018
1
Counting trees in supersymmetric quantum mechanics
Clay
Córdova
Harvard University, Cambridge, USA
Shu-Heng
Shao
Harvard University, Cambridge, USA
Supersymmetry, quiver representation, graph theory
We study the supersymmetric ground states of the Kronecker model of quiver quantum mechanics. This is the simplest quiver with two gauge groups and bifundamental matter fields, and appears universally in four-dimensional $\mathcal N = 2$ systems. The ground state degeneracy may be written as a multi-dimensional contour integral, and the enumeration of poles can be simply phrased as counting bipartite trees. We solve this combinatorics problem, thereby obtaining exact formulas for the degeneracies of an in finite class of models. We also develop an algorithm to compute the angular momentum of the ground states, and present explicit expressions for the re fined indices of theories where one rank is small.
Quantum theory
Combinatorics
1
60
10.4171/AIHPD/47
http://www.ems-ph.org/doi/10.4171/AIHPD/47
2
1
2018
A combinatorial Hopf algebra for the boson normal ordering problem
Imad Eddine
Bousbaa
USTHB, Alger, Algeria
Ali
Chouria
Université de Rouen, Saint-Étienne-du-Rouvray, France
Jean-Gabriel
Luque
Université de Rouen, Saint-Étienne-du-Rouvray, France
Normal boson ordering, Fock space, generalized Stirling numbers, combinatorial Hopf algebras
In the aim of understand the generalization of Stirling numbers occurring in the bosonic normal ordering problem, several combinatorial models have been proposed. In particular, Blasiak \emph{et al.} defined combinatorial objects allowing to interpret the number of $S_{\bf{r,s}}(k)$ appearing in the identity $(a^\dagger)^{r_n}a^{s_n}\cdots(a^\dagger)^{r_1}a^{s_1}=(a^\dagger)^\alpha\displaystyle\sum S_{\bf{r,s}}(k)(a^\dagger)^k a^k$, where $\alpha$ is assumed to be non-negative. These objects are used to define a combinatorial Hopf algebra which projects to the enveloping algebra of the Heisenberg Lie algebra. Here, we propose a new variant this construction which admits a realization with variables. This means that we construct our algebra from a free algebra $\mathbb{C}\langle A \rangle$ using quotient and shifted product. The combinatorial objects (B-diagrams) are slightly different from those proposed by Blasiak \emph{et al.}, but give also a combinatorial interpretation of the generalized Stirling numbers together with a combinatorial Hopf algebra related to Heisenberg Lie algebra. the main difference comes the fact that the B-diagrams have the same number of inputs and outputs. After studying the combinatorics and the enumeration of B-diagrams, we propose two constructions of algebras called. The Fusion algebra $\mathcal{F}$ defined using formal variables and another algebra $\mathcal{B}$ constructed directly from the B-diagrams. We show the connection between these two algebras and that $\mathcal{B}$ can be endowed with Hopf structure. We recognise two already known combinatorial Hopf subalgebras of $\mathcal{B}$: WSym the algebra of word symmetric functions indexed by set partitions and BWSym the algebra of biword symmetric functions indexed by set partitions into lists.
Combinatorics
Associative rings and algebras
Quantum theory
61
102
10.4171/AIHPD/48
http://www.ems-ph.org/doi/10.4171/AIHPD/48
2
1
2018
Right-handed Hopf algebras and the preLie forest formula
Frédéric
Menous
Université Paris-Sud, Orsay, France
Frédéric
Patras
Université de Nice, France
Forest formula, Zimmermann forest formula, preLie algebra, enveloping algebra, Hopf algebra, right-sided bialgebra
Three equivalent methods allow to compute the antipode of the Hopf algebras of Feynman diagrams in perturbative quantum field theory (QFT): the Dyson–Salam formula, the Bogoliubov formula, and the Zimmermann forest formula. Whereas the first two hold generally for arbitrary connected graded Hopf algebras, the third one requires extra structure properties of the underlying Hopf algebra but has the nice property to reduce drastically the number of terms in the expression of the antipode (it is optimal in that sense). The present article is concerned with the forest formula: we show that it generalizes to arbitrary right-handed polynomial Hopf algebras. These Hopf algebras are dual to the enveloping algebras of preLie algebras – a structure common to many combinatorial Hopf algebras which is carried in particular by the Hopf algebras of Feynman diagrams.
Associative rings and algebras
103
125
10.4171/AIHPD/49
http://www.ems-ph.org/doi/10.4171/AIHPD/49
2
1
2018
Continuous multi-line queues and TASEP
Erik
Aas
KTH - Royal Institute of Technology, Stockholm, Sweden
Svante
Linusson
KTH - Royal Institute of Technology, Stockholm, Sweden
TASEP, exclusion process, multiline queue
In this paper, we study a distribution $\Xi$ of labeled particles on a continuous ring. It arises in three di fferent ways, all related to the multi-type TASEP on a ring. We prove formulas for the probability density function for some permutations and give conjectures for a larger class. We give a complete conjecture for the probability of two particles $i, j$ being next to each other on the cycle, for which we prove some cases. We also find that two natural events associated to the process have exactly the same probability expressed as a Vandermonde determinant. It is unclear whether this is just a coincidence or a consequence of a deeper connection.
Probability theory and stochastic processes
Statistical mechanics, structure of matter
127
152
10.4171/AIHPD/50
http://www.ems-ph.org/doi/10.4171/AIHPD/50
2
1
2018
2
Perfect and separating hash families: new bounds via the algorithmic cluster expansion local lemma
Aldo
Procacci
Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
Remy
Sanchis
Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
Hash families, algorithmic Lovász local lemma, hard-core lattice gas
We present new lower bounds for the size of perfect and separating hash families ensuring their existence. Such new bounds are based on the algorithmic cluster expansion improved version of the Lovász local lemma, which also implies that the Moser–Tardos algorithm finds such hash families in polynomial time.
Combinatorics
Computer science
Statistical mechanics, structure of matter
Information and communication, circuits
153
171
10.4171/AIHPD/51
http://www.ems-ph.org/doi/10.4171/AIHPD/51
2
9
2018
Flowing to the continuum limit in tensor models for quantum gravity
Astrid
Eichhorn
University of Heidelberg, Germany
Tim
Koslowski
Universidad Nacional Autónoma de México, Ciudad de México, Mexico
Quantum gravity, functional Renormalization Group, tensor model
Tensor models provide a way to access the path-integral for discretized quantum gravity in $d$ dimensions. As in the case of matrix models for 2-dimensional quantum gravity, the continuum limit can be related to a Renormalization Group fixed point in a setup where the tensor size $N$ serves as the Renormalization Group scale. We develop functional Renormalization Group tools for tensor models with a main focus on a rank-3 model for 3-dimensional quantum gravity. We rediscover the double-scaling limit and provide an estimate for the scaling exponent. Moreover, we identify two additional fixed points with a second relevant direction in a truncation of the Renormalization Group flow. The new relevant direction might hint at the presence of additional degrees of freedom in the corresponding continuum limit.
Quantum theory
Relativity and gravitational theory
173
210
10.4171/AIHPD/52
http://www.ems-ph.org/doi/10.4171/AIHPD/52
6
4
2018
Belief propagation on replica symmetric random factor graph models
Amin
Coja-Oghlan
Goethe University, Frankfurt, Germany
Will
Perkins
University of Birmingham, UK
Random graphs, Gibbs measures, belief propagation, Bethe formula, cavity method
According to physics predictions, the free energy of random factor graph models that satisfy a certain “static replica symmetry” condition can be calculated via the Belief Propagation message passing scheme [Krzakala et al., PNAS 2007]. Here we prove this conjecture for two general classes of random factor graph models, namely Poisson random factor graphs and random regular factor graphs. Specifically, we show that the messages constructed just as in the case of acyclic factor graphs asymptotically satisfy the Belief Propagation equations and that the free energy density is given by the Bethe free energy formula.
Combinatorics
Statistical mechanics, structure of matter
211
249
10.4171/AIHPD/53
http://www.ems-ph.org/doi/10.4171/AIHPD/53
6
4
2018
A symmetry breaking transition in the edge/triangle network model
Charles
Radin
University of Texas, Austin, USA
Kui
Ren
University of Texas, Austin, USA
Lorenzo
Sadun
University of Texas, Austin, USA
Graph limits, entropy, bipodal structure, phase transitions, symmetry breaking
Our general subject is the emergence of phases, and phase transitions, in large networks subjected to a few variable constraints. Our main result is the analysis, in the model using edge and triangle subdensities for constraints, of a sharp transition between two phases with different symmetries, analogous to the transition between a fluid and a crystalline solid.
Combinatorics
Statistical mechanics, structure of matter
251
286
10.4171/AIHPD/54
http://www.ems-ph.org/doi/10.4171/AIHPD/54
6
4
2018
Phases in large combinatorial systems
Charles
Radin
University of Texas, Austin, USA
Extremal combinatorics, emergent phases
This is a status report on a companion subject to extremal combinatorics, obtained by replacing extremality properties with emergent structure, ‘phases’. We discuss phases, and phase transitions, in large graphs and large permutations, motivating and using the asymptotic formalisms of graphons for graphs and permutons for permutations. Phase structure is shown to emerge using entropy and large deviation techniques.
Combinatorics
Statistical mechanics, structure of matter
287
308
10.4171/AIHPD/55
http://www.ems-ph.org/doi/10.4171/AIHPD/55
6
4
2018
3
Horn's problem and Harish-Chandra's integrals. Probability density functions
Jean-Bernard
Zuber
Université Pierre et Marie Curie Paris 6, France
Horn problem, Harish-Chandra integrals
Horn's problem – to find the support of the spectrum of eigenvalues of the sum $C=A+B$ of two $n$ by $n$ Hermitian matrices whose eigenvalues are known – has been solved by Klyachko and by Knutson and Tao. Here the probability distribution function (PDF) of the eigenvalues of $C$ is explicitly computed for low values of $n$, for $A$ and $B$ uniformly and independently distributed on their orbit, and confronted to numerical experiments. Similar considerations apply to skew-symmetric and symmetric real matrices under the action of the orthogonal group. In the latter case, where no analytic formula is known in general and we rely on numerical experiments, curious patterns of enhancement appear.
Linear and multilinear algebra; matrix theory
Probability theory and stochastic processes
309
338
10.4171/AIHPD/56
http://www.ems-ph.org/doi/10.4171/AIHPD/56
7
25
2018
From orbital measures to Littlewood–Richardson coefficients and hive polytopes
Robert
Coquereaux
Aix-Marseille Université, Marseille, France
Jean-Bernard
Zuber
Université Pierre et Marie Curie Paris 6, France
Horn problem, honeycombs, polytopes, SU$(n)$ Littlewood–Richardson coefficients
The volume of the hive polytope (or polytope of honeycombs) associated with a Littlewood–Richardson coefficient of SU$(n)$, or with a given admissible triple of highest weights, is expressed, in the generic case, in terms of the Fourier transform of a convolution product of orbital measures. Several properties of this function – a function of three non-necessarily integral weights or of three multiplets of real eigenvalues for the associated Horn problem – are already known. In the integral case it can be thought of as a semi-classical approximation of Littlewood–Richardson coefficients. We prove that it may be expressed as a local average of a finite number of such coefficients. We also relate this function to the Littlewood–Richardson polynomials (stretching polynomials) i.e. to the Ehrhart polynomials of the relevant hive polytopes. Several SU$(n)$ examples, for $(n) = 2, 3,…, 6$, are explicitly worked out.
Nonassociative rings and algebras
Topological groups, Lie groups
Abstract harmonic analysis
Convex and discrete geometry
339
386
10.4171/AIHPD/57
http://www.ems-ph.org/doi/10.4171/AIHPD/57
7
25
2018
Collapse transition of the interacting prudent walk
Nicolas
Pétrélis
Université de Nantes, France
Niccolò
Torri
Université de Nantes, France
Polymer collapse, phase transition, prudent walk, self-avoiding random walk, free energy
This article is dedicated to the study of the 2-dimensional interacting prudent self-avoiding walk (referred to by the acronym IPSAW) and in particular to its collapse transition. The interaction intensity is denoted by $\beta > 0$ and the set of trajectories consists of those self-avoiding paths respecting the prudent condition, which means that they do not take a step towards a previously visited lattice site. The IPSAW interpolates between the interacting partially directed self-avoiding walk (IPDSAW) that was analyzed in details in, e.g., [16], [4], [5] and [10], and the interacting self-avoiding walk (ISAW) for which the collapse transition was conjectured in [11]. Three main theorems are proven. We show first that IPSAW undergoes a collapse transition at finite temperature and, up to our knowledge, there was so far no proof in the literature of the existence of a collapse transition for a non-directed model built with self-avoiding path. We also prove that the free energy of IPSAW is equal to that of a restricted version of IPSAW, i.e., the interacting two-sided prudent walk. Such free energy is computed by considering only those prudent path with a general north-east orientation. As a by-product of this result we obtain that the exponential growth rate of generic prudent paths equals that of two-sided prudent paths and this answers an open problem raised in e.g., [3] or [8]. Finally we show that, for every $\beta > 0$, the free energy of ISAW itself is always larger than $\beta$ and this rules out a possible self-touching saturation of ISAW in its conjectured collapsed phase.
Statistical mechanics, structure of matter
Combinatorics
Probability theory and stochastic processes
387
435
10.4171/AIHPD/58
http://www.ems-ph.org/doi/10.4171/AIHPD/58
7
25
2018
Moments of quantum Lévy areas using sticky shuffle Hopf algebras
Robin
Hudson
Loughborough University, UK
Uwe
Schauz
Xi’an Jiaotong – Liverpool University, Suzhou, China
Yue
Wu
Technische Universität Berlin, Germany
Lévy area, non-Fock quantum stochastic calculus, moments, sticky shuffles, Euler numbers
We study a family of quantum analogs of Lévy's stochastic area for planar Brownian motion depending on a variance parameter $\sigma \geq 1$ which deform to the classical Lévy area as $\sigma \rightarrow \infty$. They are defined as second rank iterated stochastic integrals against the components of planar Brownian motion, which are one-dimensional Brownian motions satisfying Heisenberg-type commutation relations. Such iterated integrals can be multiplied using the sticky shuffle product determined by the underlying Itô algebra of stochastic differentials. We use the corresponding Hopf algebra structure to evaluate the moments of the quantum Lévy areas and study how they deform to their classical values, which are well known to be given essentially by the Euler numbers, in the infinite variance limit.
Quantum theory
Functional analysis
437
466
10.4171/AIHPD/59
http://www.ems-ph.org/doi/10.4171/AIHPD/59
7
25
2018
4
On a causal quantum stochastic double product integral related to Lévy area
Robin
Hudson
University of Loughborough, UK
Yuchen
Pei
KTH - Royal Institute of Technology, Stockholm, Sweden
causal double product, Lévy's stochastic area, position and momentum Brownian motions, linear extensions, Catalan numbers, Dyck paths
We study the family of causal double product integrals \begin{equation*} \prod_{a < x < y < b}\Big(1 + i{\lambda \over 2}(dP_x dQ_y - dQ_x dP_y) + i {\mu \over 2}(dP_x dP_y + dQ_x dQ_y)\Big), \end{equation*} where $P$ and $Q$ are the mutually noncommuting momentum and position Brownian motions of quantum stochastic calculus. The evaluation is motivated heuristically by approximating the continuous double product by a discrete product in which infinitesimals are replaced by finite increments. The latter is in turn approximated by the second quantisation of a discrete double product of rotation-like operators in different planes due to a result in [15]. The main problem solved in this paper is the explicit evaluation of the continuum limit $W$ of the latter, and showing that $W$ is a unitary operator. The kernel of $W - I$ is written in terms of Bessel functions, and the evaluation is achieved by working on a lattice path model and enumerating linear extensions of related partial orderings, where the enumeration turns out to be heavily related to Dyck paths and generalisations of Catalan numbers.
Quantum theory
Combinatorics
Order, lattices, ordered algebraic structures
467
512
10.4171/AIHPD/60
http://www.ems-ph.org/doi/10.4171/AIHPD/60
7
25
2018
Rigged configurations and cylindric loop Schur functions
Thomas
Lam
University of Michigan, Ann Arbor, USA
Pavlo
Pylyavskyy
University of Minnesota, Minneapolis, USA
Reiho
Sakamoto
Tokyo University of Science, Japan
Rigged configuration, discrete soliton, box ball system, tropicalization, loop schur functions
Rigged configurations are known to provide action-angle variables for remarkable discrete dynamical systems known as box-ball systems. We conjecture an explicit piecewise-linear formula to obtain the shapes of a rigged configuration from a tensor product of one-row crystals. We introduce cylindric loop Schur functions and show that they are invariants of the geometric $R$-matrix. Our piecewise-linear formula is obtained as the tropicalization of ratios of cylindric loop Schur functions. We prove our conjecture for the first shape of a rigged configuration, thus giving a piecewise-linear formula for the lengths of the solitons of a box-ball system.
Combinatorics
Dynamical systems and ergodic theory
Mechanics of particles and systems
513
555
10.4171/AIHPD/61
http://www.ems-ph.org/doi/10.4171/AIHPD/61
7
25
2018
Edge correlation function of the 8-vertex model when $a + c = b + d$
Jérôme
Casse
Université de Lorraine, Vandoeuvre-lès-Nancy, France
8-vertex model, correlation function, system of particles, probabilistic cellular automata
This paper is devoted to the 8-vertex model and its edge correlation function. In some particular (integrable) cases, we find a closed form of the edge correlation function and we deduce also its asymptotics. In addition, we quantify influence of boundary conditions on this function. To do this, we introduce a system of particles in interaction related to the 8-vertex model. This system, studied using various tools fromanalytic combinatorics, random walks and conics, permits to compute the correlation function. To study the influence of boundary conditions, we involve probabilistic cellular automata of order 2.
Statistical mechanics, structure of matter
Combinatorics
Probability theory and stochastic processes
557
619
10.4171/AIHPD/63
http://www.ems-ph.org/doi/10.4171/AIHPD/63
7
25
2018