Discrete Torus Research Articles

On (global) unique continuation properties of the fractional discrete Laplacian

We study various qualitative and quantitative (global) unique continuation properties for the fractional discrete Laplacian. We show that while the fractional Laplacian enjoys striking rigidity properties in the form of (global) unique continuation properties, the fractional discrete Laplacian does not enjoy these in general. While discretization thus counteracts the strong rigidity properties of the continuum fractional Laplacian, by discussing quantitative forms of unique continuation, we illustrate that these properties can be recovered if exponentially small (in the lattice size) correction terms are added. In particular, this allows us to deduce stability properties for a discrete, linear inverse problem for the fractional Laplacian. We complement these observations with a transference principle and the discussion of these properties on the discrete torus.

Spatio–Spectral Limiting on Replacements of Tori by Cubes

A class of graphs is defined in which each vertex of a discrete torus is replaced by a Boolean hypercube in such a way that vertices in a fixed subset of each replacement cube are adjacent to corresponding vertices of a neighboring replacement cube. Bases of eigenvectors of the Laplacians of the resulting graphs are described in a manner suitable for quantifying the concentration of a low-spectrum vertex function on a single vertex replacement. Functions that optimize this concentration on these graphs can be regarded as analogues of Slepian prolate functions that optimize concentration of a bandlimited signal on an interval in the classical setting of the real line. Comparison to the case of a simple discrete cycle shows that replacement allows for higher concentration.

Weights for ℓ-local compact groups

In this note, we initiate the study of F-weights for an ℓ-local compact group F over a discrete ℓ-toral group S with discrete torus T. Motivated by Alperin's Weight Conjecture for simple groups of Lie-type, we conjecture that when T is the unique maximal abelian subgroup of S up to F-conjugacy and every element of S is F-fused into T, the number of weights of F is bounded above by the number of ordinary irreducible characters of its Weyl group. By combining the structure theory of F with the theory of blocks with cyclic defect group, we are able to give a proof of this conjecture in the case when F is simple and |S:T|=ℓ. We also propose and give evidence for an analogue of the height zero case of Robinson's Ordinary Weight conjecture in this setting.

Homomorphisms from the torus

We present a detailed probabilistic and structural analysis of the set of weighted homomorphisms from the discrete torus Zmn, where m is even, to any fixed graph: we show that the corresponding probability distribution on such homomorphisms is close to a distribution defined constructively as a certain random perturbation of some dominant phase. This has several consequences, including solutions (in a strong form) to conjectures of Engbers and Galvin and a conjecture of Kahn and Park. Special cases include sharp asymptotics for the number of independent sets and the number of proper q-colourings of Zmn (so in particular, the discrete hypercube). We give further applications to the study of height functions and (generalised) rank functions on the discrete hypercube and disprove a conjecture of Kahn and Lawrenz. For the proof we combine methods from statistical physics, entropy and graph containers and exploit isoperimetric and algebraic properties of the torus.

Stochastic Homogenization of Gaussian Fields on Random Media

In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields Ξg,a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Xi ^{g,\ extbf{a}} $$\\end{document} and bi-Laplacian fields Ξb,a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Xi ^{b,\ extbf{a}}$$\\end{document}. They can be characterized as follows: for f=δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f=\\delta $$\\end{document} the solution u of ∇·a∇u=f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ abla \\cdot \ extbf{a} \ abla u =f$$\\end{document}, a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{a}$$\\end{document} is a uniformly elliptic random environment, is the covariance of Ξg,a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Xi ^{g,\ extbf{a}}$$\\end{document}. When f is the white noise, the field Ξb,a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Xi ^{b,\ extbf{a}}$$\\end{document} can be viewed as the distributional solution of the same elliptic equation. Our results characterize the scaling limit of such fields on both, a sufficiently regular domain D⊂Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D\\subset \\mathbb {R}^d$$\\end{document}, or on the discrete torus. Based on stochastic homogenization techniques applied to the eigenfunction basis of the Laplace operator Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document}, we will show that such families of fields converge to an appropriate multiple of the GFF resp. bi-Laplacian. The limiting fields are determined by their respective homogenized operator a¯Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\\mathrm{\\bar{\ extbf{a}}}\\,}}\\Delta $$\\end{document}, with constant a¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\\mathrm{\\bar{\ extbf{a}}}\\,}}$$\\end{document} depending on the law of the environment a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{a}$$\\end{document}. The proofs are based on the results found in Armstrong et al. (in: Grundlehren der mathematischen Wissenschaften, Springer International Publishing, Cham, 2019) and Gloria et al. (ESAIM Math Model Numer Anal 48(2):325-346, 2014).

High-dimensional near-critical percolation and the torus plateau

We consider percolation on Zd and on the d-dimensional discrete torus, in dimensions d≥11 for the nearest-neighbour model and in dimensions d>6 for spread-out models. For Zd we employ a wide range of techniques and previous results to prove that there exist positive constants c and C such that the slightly subcritical two-point function and one-arm probabilities satisfy Ppc−ε(0↔x)≤C‖x‖d−2e−cε1/2‖x‖,cr2e−Cε1/2r≤Ppc−ε(0↔∂[−r,r]d)≤Cr2e−cε1/2r. Using this, we prove that throughout the critical window the torus two-point function has a “plateau,” meaning that it decays for small x as ‖x‖−(d−2) but for large x is essentially constant and of order V−2/3 where V is the volume of the torus. The plateau for the two-point function leads immediately to a proof of the torus triangle condition, which is known to have many implications for the critical behaviour on the torus, and also leads to a proof that the critical values on the torus and on Zd are separated by a multiple of V−1/3. The torus triangle condition and the size of the separation of critical points have been proved previously, but our proofs are different and are direct consequences of the bound on the Zd two-point function. In particular, we use results derived from the lace expansion on Zd, but in contrast to previous work on high-dimensional torus percolation, we do not need or use a separate torus lace expansion.

The Near-Critical Two-Point Function and the Torus Plateau for Weakly Self-avoiding Walk in High Dimensions

We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice $$\mathbb {Z}^d$$ in dimensions $$d>4$$ , in the vicinity of the critical point, and prove an upper bound $$|x|^{-(d-2)}\exp [-c|x|/\xi ]$$ , where the correlation length $$\xi $$ has a square root divergence at the critical point. As an application, we prove that the two-point function for weakly self-avoiding walk on a discrete torus in dimensions $$d{>}4$$ has a “plateau.” We also discuss the significance and consequences of the plateau for the analysis of critical behaviour on the torus.

Counting rooted spanning forests for circulant foliation over a graph

In this paper, we present a new method to produce explicit formulas for the number of rooted spanning forests $f(n)$ for the infinite family of graphs $H_n=H_n(G_1,G_2,\ldots,G_m)$ obtained as a circulant foliation over a graph $H$ on $m$ vertices with fibers $G_1,G_2,\ldots,G_m.$ Each fiber $G_i=C_n(s_{i,1},s_{i,2},\ldots,s_{i,k_i})$ of this foliation is the circulant graph on $n$ vertices with jumps $s_{i,1},s_{i,2},\ldots,s_{i,k_i}.$ This family includes the family of generalized Petersen graphs, $I$-graphs, sandwiches of circulant graphs, discrete torus graphs and others. The formulas are expressed through Chebyshev polynomials. We prove that the number of rooted spanning forests can be represented in the form $f(n)=p f(H)a(n)^2,$ where $a(n)$ is an integer sequence and $p$ is a prescribed natural number depending on the number of odd elements in the set of $s_{i,j}.$ Finally, we find an asymptotic formula for $f(n)$ through the Mahler measure of the associated Laurent polynomial.

Crux and Long Cycles in Graphs

We introduce a notion of the crux of a graph $G$, measuring the order of a smallest dense subgraph in $G$. This simple-looking notion leads to some generalizations of known results about cycles, offering an interesting paradigm of “replacing average degree by crux.” In particular, we prove that every graph contains a cycle of length linear in its crux. Long proved that every subgraph of a hypercube $Q^m$ (resp., discrete torus $C_3^m$) with average degree $d$ contains a path of length $2^{d/2}$ (resp., $2^{d/4}$) and conjectured that there should be a path of length $2^{d}-1$ (resp., $3^{d/2}-1$). As a corollary of our result, together with isoperimetric inequalities, we close these exponential gaps giving asymptotically optimal bounds on long paths in hypercubes, discrete tori, and more generally Hamming graphs. We also consider random subgraphs of $C_4$-free graphs and hypercubes, proving near optimal lower bounds on the lengths of long cycles.

On an approach for evaluating certain trigonometric character sums using the discrete time heat kernel

In this article we develop a general method by which one can explicitly evaluate certain sums of nth powers of products of d≥1 elementary trigonometric functions evaluated at m=(m1,…,md)-th roots of unity. Our approach is to first identify the individual terms in the expression under consideration as eigenvalues of a discrete Laplace operator associated to a graph whose vertices form a d-dimensional discrete torus Gm which depends on m. The sums in question are then related to the nth step of a Markov chain on Gm. The Markov chain admits the interpretation as a particular random walk, also viewed as a discrete time and discrete space heat diffusion, so then the sum in question is related to special values of the associated heat kernel. Our evaluation follows by deriving a combinatorial expression for the heat kernel, which is obtained by periodizing the heat kernel on the infinite lattice Zd which covers Gm.

Coalescing and branching simple symmetric exclusion process

Motivated by kinetically constrained interacting particle systems (KCM), we consider a reversible coalescing and branching simple exclusion process on a general finite graph G=(V,E) dual to the biased voter model on G. Our main goal is tight bounds on its logarithmic Sobolev constant and relaxation time, with particular focus on the delicate slightly supercritical regime in which the equilibrium density of particles tends to zero as |V|→∞. Our results allow us to recover very directly and improve to ℓp-mixing, p≥2, and to more general graphs, the mixing time results of Pillai and Smith for the Fredrickson–Andersen one spin facilitated (FA-1f) KCM on the discrete d-dimensional torus. In view of applications to the more complex FA-jf KCM, j>1, we also extend part of the analysis to an analogous process with a more general product state space.

Geometric dominating sets - a minimum version of the No-Three-In-Line Problem

We consider a minimizing variant of the well-known No-Three-In-Line Problem, the Geometric Dominating Set Problem: What is the smallest number of points in an n×n grid such that every grid point lies on a common line with two of the points in the set? We show a lower bound of Ω(n2/3) points and provide a constructive upper bound of size 2⌈n/2⌉. If the points of the dominating sets are required to be in general position we provide optimal solutions for grids of size up to 12×12. For arbitrary n the currently best upper bound for points in general position remains the obvious 2n. Finally, we discuss the problem on the discrete torus where we prove an upper bound of O((nlog⁡n)1/2). For n even or a multiple of 3, we can even show a constant upper bound of 4. We also mention a number of open questions and some further variations of the problem.

Large deviations in the supremum norm for a reaction–diffusion system

We present large deviations estimates in the supremum norm for a system of independent random walks superposed with a birth-and-death dynamics evolving on the discrete torus with N sites. The scaling limit considered is the so-called high density limit (see the survey in (In From Particle Systems to Partial Differential Equations (2014) 179–189 Springer) on the subject), where space, time and initial quantity of particles are rescaled. The associated rate functional here obtained is a semi-linearized version of the rate function of (Probability Theory and Related Fields 97 (1992) 339–361), which dealt with large deviations of exclusion processes superposed with birth-and-death dynamics. An important ingredient in the proof of large deviations consists in providing a limit of a suitable class of perturbations of the original process, which is precisely one of the main contributions of this work: a strategy to extend the original high density approach (as in Advances in Applied Probability 12 (1980) 367–379; Annals of Probability 19 (1991) 1440–1462; The Annals of Applied Probability 2 (1992) 131–141; Journal of Statistical Physics 149 (2012) 629–642; Annals of Probability 14 (1986) 173–193; Probability Theory and Related Fields 78 (1988) 11–37) to weakly asymmetric systems. Two cases are considered with respect to the initial quantity of particles, the power law and the (at least) exponential growth. In the first case, we present the lower bound only on a certain set of smooth profiles, while in the second case under some extra technical assumptions we provide a full large deviations principle.

Concentration inequalities for log-concave distributions with applications to random surface fluctuations

We derive two concentration inequalities for linear functions of log-concave distributions: an enhanced version of the classical Brascamp–Lieb concentration inequality and an inequality quantifying log-concavity of marginals in a manner suitable for obtaining variance and tail probability bounds. These inequalities are applied to the statistical mechanics problem of estimating the fluctuations of random surfaces of the ∇φ type. The classical Brascamp–Lieb inequality bounds the fluctuations whenever the interaction potential is uniformly convex. We extend these bounds to the case of convex potentials whose second derivative vanishes only on a zero measure set, when the underlying graph is a d-dimensional discrete torus. The result applies, in particular, to potentials of the form U(x)=|x|p with p>1 and answers a question discussed by Brascamp–Lieb–Lebowitz (In Statistical Mechanics (1975) 379–390, Springer). Additionally, new tail probability bounds are obtained for the family of potentials U(x)=|x|p+x2, p>2. This result answers a question mentioned by Deuschel and Giacomin (Stochastic Process. Appl. 89 (2000) 333–354).

Constructing fractional Gaussian fields from long-range divisible sandpiles on the torus

In Cipriani et al. (2017), the authors proved that, with the appropriate rescaling, the odometer of the (nearest neighbours) divisible sandpile on the unit torus converges to a bi-Laplacian field. Here, we study α-long-range divisible sandpiles, similar to those introduced in Frómeta and Jara (2018). We show that, for α∈(0,2), the limiting field is a fractional Gaussian field on the torus with parameter α/2. However, for α∈[2,∞), we recover the bi-Laplacian field. This provides an alternative construction of fractional Gaussian fields such as the Gaussian Free Field or membrane model using a diffusion based on the generator of Lévy walks. The central tool for obtaining our results is a careful study of the spectrum of the fractional Laplacian on the discrete torus. More specifically, we need the rate of divergence of the eigenvalues as we let the side length of the discrete torus go to infinity. As a side result, we obtain precise asymptotics for the eigenvalues of discrete fractional Laplacians. Furthermore, we determine the order of the expected maximum of the discrete fractional Gaussian field with parameter γ=min{α,2} and α∈R+∖{2} on a finite grid.

Self-organizing Strategies in Game of Agent Movement

In this paper, a stochastic game model of self-organization of strategies of stochastic game of mobile agents in the form of cyclic behavioral patterns, which consist of coordinated strategies for moving agents in a limited discrete space, is developed. The behavioral pattern of a multi-agent system is a visualized form of orderly movement of agents that arises from their initial chaotic movement during the learning of a stochastic game. The mobility of multi-step stochastic game agents is ensured by the fact that in discrete moments of time they randomly, simultaneously and independently choose their own pure strategy of moving in one of the possible directions. Current player payments are defined as loss functions that depend on the strategies of neighboring players. These functions are formed from the penalty for irregular spacing of agents in a limited discrete space and the penalty for collisions when moving agents. Random selection of players’ strategies aims to minimize their average loss functions. The generation of sequences of pure strategies is performed by a discrete distribution based on the vectors of mixed strategies. The elements of the vectors of mixed strategies are the conditional probabilities of choosing the appropriate pure displacement strategies. Mixed strategies change over time, adaptively taking into account the value of current losses. This provides an increase in the probability of choosing those strategies that lead to a decrease in the functions of average losses. The dynamics of the vectors of mixed strategies is determined by the Markov recurrent method, for the construction of which a stochastic approximation of the modified condition of complementary non- rigidity, which is valid at Nash equilibrium points, is performed, and a projection operator for expandable unit epsilon simplex is applied. The convergence of the recurrent game method is ensured by compliance with the fundamental conditions and limitations of stochastic approximation. The stochastic game begins with untrained mixed strategies that set a chaotic picture of moving agents. During the learning of the stochastic game, the vectors of mixed strategies are purposefully changed so as to ensure an orderly, conflict-free movement of agents. As a result of computer simulation of stochastic game, cyclic patterns of self-organization of mobile agents on the surface of a discrete torus and within a rectangular region on a plane are obtained. The reliability of the experimental studies was confirmed by the similarity of the obtained patterns of moving agents for different sequences of random variables. The results of the study are proposed to be used in practice for the construction of distributed systems with elements of self-organization, solving various flow and transport problems and collective decision-making in conditions of uncertainty.

NISQ circuit compilation is the travelling salesman problem on a torus

Noisy, intermediate-scale quantum (NISQ) computers are expected to execute quantum circuits of up to a few hundred qubits. The circuits have to conform to NISQ architectural constraints regarding qubit allocation and the execution of multi-qubit gates. Quantum circuit compilation (QCC) takes a nonconforming circuit and outputs a compatible circuit. Can classical optimisation methods be used for QCC? Compilation is a known combinatorial problem shown to be solvable by two types of operations: (1) qubit allocation, and (2) gate scheduling. We show informally that the two operations form a discrete ring. The search landscape of QCC is a two dimensional discrete torus where vertices represent configurations of how circuit qubits are allocated to NISQ registers. Torus edges are weighted by the cost of scheduling circuit gates. The novelty of our approach uses the fact that a circuit’s gate list is circular: compilation can start from any gate as long as all the gates will be processed, and the compiled circuit has the correct gate order. Our work bridges a theoretical and practical gap between classical circuit design automation and the emerging field of quantum circuit optimisation.

Boolean Functions: Noise Stability, Non-Interactive Correlation Distillation, and Mutual Information

Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$T_{\epsilon }$ </tex-math></inline-formula> be the noise operator acting on Boolean functions <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f:\{0, 1\}^{n}\to \{0, 1\}$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon \in [{0, 1/2}]$ </tex-math></inline-formula> is the noise parameter. Given <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha &gt;1$ </tex-math></inline-formula> and fixed mean <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {E} f$ </tex-math></inline-formula> , which Boolean function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> has the largest <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -th moment <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {E}(T_\epsilon f)^\alpha $ </tex-math></inline-formula> ? This question has close connections with noise stability of Boolean functions, the problem of non-interactive correlation distillation, and Courtade-Kumar’s conjecture on the most informative Boolean function. In this paper, we characterize maximizers in some extremal settings, such as low noise ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon =\epsilon (n)$ </tex-math></inline-formula> close to 0), high noise ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon =\epsilon (n)$ </tex-math></inline-formula> close to 1/2), as well as when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha =\alpha (n)$ </tex-math></inline-formula> is large. Analogous results are also established in more general contexts, such as Boolean functions defined on discrete torus <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\mathbb {Z}/p \mathbb {Z})^{n}$ </tex-math></inline-formula> and the problem of noise stability in a tree model.

Search on vertex-transitive graphs by lackadaisical quantum walk

The lackadaisical quantum walk is a discrete-time, coined quantum walk on a graph with a weighted self-loop at each vertex. It uses a generalized Grover coin and the flip-flop shift, which makes it equivalent to Szegedy's quantum Markov chain. It has been shown that a lackadaisical quantum walk can improve spatial search on the complete graph, discrete torus, cycle, and regular complete bipartite graph. In this paper, we observe that these are all vertex-transitive graphs, and when there is a unique marked vertex, the optimal weight of the self-loop equals the degree of the loopless graph divided by the total number of vertices. We propose that this holds for all vertex-transitive graphs with a unique marked vertex. We present a number of numerical simulations supporting this hypothesis, including search on periodic cubic lattices of arbitrary dimension, strongly regular graphs, Johnson graphs, and the hypercube.

DISCRETE CALCULUS WITH CUBIC CELLS ON DISCRETE MANIFOLDS

This work is thought as an operative guide to discrete exterior calculus (DEC), but at the same time with a rigorous exposition. We present a version of (DEC) on cubic cell, defining it for discrete manifolds. An example of how it works, it is done on the discrete torus, where usual Gauss and Stokes theorems are recovered.