Providing performance guarantees is one of the critical objectives of recent and future communication networks, toward which regulations, i.e., constraints on key system parameters, have played an indispensable role. This is the case for large wireless communication networks, where spatial regulations (e.g., constraints on intercell distance) have recently been shown, through a spatial network calculus, to be essential for establishing provable wireless link-level guarantees. In this work, we focus on performance guarantees for the downlink of cellular networks where we impose a hardcore (spatial) regulation on base station (BS) locations and evaluate how BS scheduling (which controls which BSs can transmit at a given time) impacts performance. Hardcore regulation is the simplest form of spatial regulation that enforces a minimal distance between any pair of transmitters in the network. Within this framework of spatial network calculus, we first provide an upper bound on the power of total interference for a spatially regulated cellular network, and then, identify the regimes where scheduling BSs yields better link-level rate guarantees compared to scenarios where base stations are
The hardcore model is one of the most classic and widely studied examples of undirected graphical models. Given a graph $G$, the hardcore model describes a Gibbs distribution of $λ$-weighted independent sets of $G$. In the last two decades, a beautiful computational phase transition has been established at a precise threshold $λ_c(Δ)$ where $Δ$ denotes the maximum degree, where the task of sampling independent sets transitions from polynomial-time solvable to computationally intractable. We study the critical hardcore model where $λ= λ_c(Δ)$ and show that the Glauber dynamics, a simple yet popular Markov chain algorithm, mixes in $\tilde{O}(n^{4+O(1/Δ)})$ time on any $n$-vertex graph of maximum degree $Δ\geq3$, significantly improving the previous upper bound $\tilde{O}(n^{12.88+O(1/Δ)})$ by the recent work arXiv:2411.03413. Our improvement comes from an optimal bound on the $\ell_\infty$-spectral independence for the hardcore model at all subcritical fugacity $λ< λ_c(Δ)$.
Neither hardcore bosons nor fermions can occupy the same lattice site-state. However, a nearest neighbour interaction may counteract the hardcore effect, resulting in condensate states in a bosonic system. In this work, we unveil the underlying mechanism by developing a general method to construct the condensate eigenstates from those of sub-Hamiltonians. As an application, we find that a local on-site potential can induce an evanescent condensate mode. Based on this, exact condensate ground states of hardcore bosons, possessing off-diagonal long-range order, can be constructed when an array of impurities is applied. The effect of the off-resonance impurity on the condensate ground states is also investigated using numerical simulations of the dynamic response.
We design an efficient sampling algorithm to generate samples from the hardcore model on random regular bipartite graphs as long as $λ\lesssim \frac{1}{\sqrtΔ}$, where $Δ$ is the degree. Combined with recent work of Jenssen, Keevash and Perkins this implies an FPRAS for the partition function of the hardcore model on random regular bipartite graphs at any fugacity. Our algorithm is shown by analyzing two new Markov chains that work in complementary regimes. Our proof then proceeds by showing the corresponding simplicial complexes are top-link spectral expanders and appealing to the trickle-down theorem to prove fast mixing.
We study the Hilbert space fragmentation (HSF) in hardcore Bose and Fermi Hubbard models in the framework of the restricted spectrum generating algebra (RSGA). We present a family of hardcore Bose-Hubbard models with repulsive density-density interactions on a generalized Lieb lattice. We show that this system possesses the RSGA structure in the large interaction strength limit, exhibiting quantum HSF. It allows us to construct a set of exact condensate eigenstates, possessing off diagonal long-range order. Based on numerical simulations conducted on several representative lattices, we demonstrate the existence of weak fragmentations when the constraints are not exact. As applications, we also studied the connection between HSF and RSGA in modified fermionic Hubbard models, where the η-pairing states are shown to be energy towers, acting as quantum scars.
Consider a function that is mildly hard for size-$s$ circuits. For sufficiently large $s$, Impagliazzo's hardcore lemma guarantees a constant-density subset of inputs on which the same function is extremely hard for circuits of size $s'<\!\!<s$. Blanc, Hayderi, Koch, and Tan [FOCS 2024] recently showed that the degradation from $s$ to $s'$ in this lemma is quantitatively tight in certain parameter regimes. We give a simpler and more general proof of this result in almost all parameter regimes of interest by showing that a random junta witnesses the tightness of the hardcore lemma with high probability.
Smooth boosters generate distributions that do not place too much weight on any given example. Originally introduced for their noise-tolerant properties, such boosters have also found applications in differential privacy, reproducibility, and quantum learning theory. We study and settle the sample complexity of smooth boosting: we exhibit a class that can be weak learned to $γ$-advantage over smooth distributions with $m$ samples, for which strong learning over the uniform distribution requires $\tildeΩ(1/γ^2)\cdot m$ samples. This matches the overhead of existing smooth boosters and provides the first separation from the setting of distribution-independent boosting, for which the corresponding overhead is $O(1/γ)$. Our work also sheds new light on Impagliazzo's hardcore theorem from complexity theory, all known proofs of which can be cast in the framework of smooth boosting. For a function $f$ that is mildly hard against size-$s$ circuits, the hardcore theorem provides a set of inputs on which $f$ is extremely hard against size-$s'$ circuits. A downside of this important result is the loss in circuit size, i.e. that $s' \ll s$. Answering a question of Trevisan, we show that this s
We extend the hardcore model to a multicoloured version: a subset of vertices of a graph are coloured such that no vertex is adjacent to one of the same colour; uncoloured vertices do not constrain neighbours. This mathematically models multi-channel resource sharing, such as fibreoptic routing. We analyse certain simple Glauber-type dynamics on such configurations, and find conditions which ensure fast mixing. These dynamics model a queueing system: customers queue for service at vertices, who only serve customers whilst they are coloured in the underlying configuration; uncoloured vertices sit idle. The mixing estimates are applied to control queue lengths in equilibrium.
Hardcore functions have been used as a technical tool to construct secure cryptographic systems; however, little is known on their quantum counterpart, called quantum hardcore functions. With a new insight into fundamental properties of quantum hardcores, we present three new quantum hardcore functions for any (strong) quantum one-way function. We also give a "quantum" solution to Damgard's question (CRYPTO'88) on a classical hardcore property of his pseudorandom generator, by proving its quantum hardcore property. Our major technical tool is the new notion of quantum list-decoding of "classical" error-correcting codes (rather than "quantum" error-correcting codes), which is defined on the platform of computational complexity theory and computational cryptography (rather than information theory). In particular, we give a simple but powerful criterion that makes a polynomial-time computable classical block code (seen as a function) a quantum hardcore for all quantum one-way functions. On their own interest, we construct efficient quantum list-decoding algorithms for classical block codes whose associated quantum states (called codeword states) form a nearly phase-orthogonal basis.
We propose a new point process model that combines, in the spatio-temporal setting, both multi-scaling by hybridization and hardcore distances. Our so-called hybrid Strauss hardcore point process model allows different types of interaction, at different spatial and/or temporal scales, that might be of interest in environmental and biological applications. The inference and simulation of the model are implemented using the logistic likelihood approach and the birth-death Metropolis-Hastings algorithm. Our model is illustrated and compared to others on two datasets of forest fire occurrences respectively in France and Spain.
We characterize the uniqueness condition in the hardcore model for bipartite graphs with degree bounds only on one side, and provide a nearly linear time sampling algorithm that works up to the uniqueness threshold. We show that the uniqueness threshold for bipartite graph has almost the same form of the tree uniqueness threshold for general graphs, except with degree bounds only on one side of the bipartition. The hardcore model from statistical physics can be seen as a weighted enumeration of independent sets. Its bipartite version (#BIS) is a central open problem in approximate counting. Compared to the same problem in a general graph, surprising tractable regime have been identified that are believed to be hard in general. This is made possible by two lines of algorithmic approach: the high-temperature algorithms starting from Liu and Lu (STOC 2015), and the low-temperature algorithms starting from Helmuth, Perkins, and Regts (STOC 2019). In this work, we study the limit of these algorithms in the high-temperature case. Our characterization of the uniqueness condition is obtained by proving decay of correlations for arguably the best possible regime, which involves locating fix
We consider the 1D Tonks-Girardeau gas with a space-dependent potential out of equilibrium. We derive the exact dynamics of the system when divided into $n$ boxes and decomposed into energy eigenstates within each box. It is a representation of the wave function that is mixed between real space and momentum space, whose basis elements are plane waves localized in a box, motivating the word "wavelet". In this representation we derive the emergence of generalized hydrodynamics in appropriate limits without assuming local relaxation. We emphasize in particular that a generalized hydrodynamic behaviour emerges in a high-momentum and short-time limit, besides the more common large-space and late-time limit, which is akin to a semi-classical expansion. In this limit, conserved charges do not require a large number of particles to be described by generalized hydrodynamics. Besides, we show that this wavelet representation provides an efficient numerical algorithm for a complete description of out-of-equilibrium dynamics of hardcore bosons.
We formulate a continuous version of the well known discrete hardcore (or independent set) model on a locally finite graph, parameterized by the so-called activity parameter $λ> 0$. In this version, the state or "spin value" $x_u$ of any node $u$ of the graph lies in the interval $[0,1]$, the hardcore constraint $x_u + x_v \leq 1$ is satisfied for every edge $(u,v)$ of the graph, and the space of feasible configurations is given by a convex polytope. When the graph is a regular tree, we show that there is a unique Gibbs measure associated to each activity parameter $λ>0$. Our result shows that, in contrast to the standard discrete hardcore model, the continuous hardcore model does not exhibit a phase transition on the infinite regular tree. We also consider a family of continuous models that interpolate between the discrete and continuous hardcore models on a regular tree when $λ= 1$ and show that each member of the family has a unique Gibbs measure, even when the discrete model does not. In each case, the proof entails the analysis of an associated Hamiltonian dynamical system that describes a certain limit of the marginal distribution at a node. Furthermore, given any seque
We study 2-flavor Hamiltonian lattice QCD in (1+1)D with hardcore gluons, at zero and finite density, by means of matrix product states. We introduce a formulation of the theory where gauge redundancy is absent and construct a gauge invariant tensor network ansatz. We show that the model is critical in an extended subregion of parameter space and identify at least two distinct phases, one of which embeds the continuum limit location. We reconstruct a subset of the particle spectrum in each phase, identifying edge and bulk gapless modes. We thereby show that the studied model provides a minimal SU(3) gauge theory whilst reproducing known phenomena of (3+1)D QCD. Most notably, it features charged pions.
In this paper we analyze synthetic mobility traces generated for three-lane unidirectional motorway traffic to find that the locations of vehicles along a lane are better modeled by a hardcore point process instead of the widely-accepted Poisson point process (PPP). In order to capture the repulsion between successive vehicles while maintaining a level of analytical tractability, we make a simple extension to PPP: We model the inter-vehicle distance along a lane equal to the sum of a constant hardcore distance and an exponentially distributed random variable. We calculate the J-function and the Ripley's K-function for this hardcore point process. We fit its parameters to the available traces, and we illustrate that the higher the average speed along a lane, the more prominent the hardcore component becomes. In addition, we consider a transmitter-receiver link on the same lane, and we generate simple formulae for the moments of interference under reduced Palm measure for that lane, and without conditioning for other lanes. We illustrate that under Rayleigh fading a shifted-gamma approximation for the distribution of interference per lane provides a very good fit to the simulated out
Dynamic correlations of quantum magnets provide valuable information. However, especially at finite temperature, many observations, e.g., by inelastic neutron scattering experiments, are not yet quantitatively understood. Generically, the elementary excitations of gapped quantum magnets are hardcore bosons because no two of them can occupy the same site. The previously introduced diagrammatic Brückner approach dealt with hardcore bosons of a single flavor at finite temperature. Here, this approach is extended to hardcore bosons of several kinds (flavors). The approach based on ladder diagrams is gauged with so far unknown rigorous results for the thermal occupation function of multiflavored hardcore bosons in one dimension with nearest-neighbor hopping. For low temperatures, it works very well. Furthermore, we study to which extent the approach is a conserving approximation. Empirical evidence shows that this is true only in the single-flavor case.
We study the many-body ground states of two-component hardcore bosons in topological triangular lattice models. Utilizing exact diagonalization and density-matrix renormalization group calculations, we demonstrate that at commensurate two-thirds filling per lattice site, two-component bosonic integer quantum hall (BIQH) effect emerges with the associated $\mathbf{K}=\begin{pmatrix} 0 & 1\\ 1 & 0\\ \end{pmatrix}$ matrix under strong intercomponent Hubbard repulsion. The topological nature is further elucidated by (i) a unique ground state degeneracy with a robust spectrum gap, (ii) a quantized topological Chern number matrix $\mathbf{C}=\mathbf{K}^{-1}$, and (iii) two counterpropagating edge branches. Moreover, with increasing nearest-neighbor repulsions, the ground state undergoes a first-order transition from a BIQH liquid to a commensurate solid order.
Hardcore and Ising models are two most important families of two state spin systems in statistic physics. Partition function of spin systems is the center concept in statistic physics which connects microscopic particles and their interactions with their macroscopic and statistical properties of materials such as energy, entropy, ferromagnetism, etc. If each local interaction of the system involves only two particles, the system can be described by a graph. In this case, fully polynomial-time approximation scheme (FPTAS) for computing the partition function of both hardcore and anti-ferromagnetic Ising model was designed up to the uniqueness condition of the system. These result are the best possible since approximately computing the partition function beyond this threshold is NP-hard. In this paper, we generalize these results to general physics systems, where each local interaction may involves multiple particles. Such systems are described by hypergraphs. For hardcore model, we also provide FPTAS up to the uniqueness condition, and for anti-ferromagnetic Ising model, we obtain FPTAS where a slightly stronger condition holds.
In this article we study a set of integrable quantum cellular automata,the quantum hardcore gases (QHCG), with an arbitrary local Hilbert space dimension, and discuss the matrix product ansatz based approach for solving the dynamics of local operators analytically. Subsequently, we focus on the dynamics of operator spreading, in particular on the out-of-time ordered correlation functions (OTOCs), operator weight spreading and operators space entanglement entropy (OSEE). All of the quantities were conjectured to provide signifying features of integrable systems and quantum chaos. We show that in QHCG OTOCs spread diffusively and that in the limit of the large local Hilbert space dimension they increase linearly with time, despite their integrability. On the other hand, it was recently conjectured that operator weight front, which is associated with the extent of operators, spreads diffusively in both, integrable and generic systems, but its decay seems to differ in these two cases. We observe that the spreading of the operator weight front in QHCG is markedly different from chaotic, generic integrable and free systems, as the front freezes in the long time limit. Finally, we discuss
We consider two strongly correlated two-component quantum systems, consisting of quantum mobile particles and classical immobile particles. The both systems are described by Falicov-Kimball-like Hamiltonians on a square lattice, extended by direct short-range interactions between the immobile particles. In the first system the mobile particles are spinless fermions while in the second one they are hardcore bosons. We construct rigorously ground-state phase diagrams of the both systems in the strong-coupling regime and at half-filling. Two main conclusions are drawn. Firstly, short-range interactions in quantum gases are sufficient for the appearance of charge stripe-ordered phases. By varying the intensity of a direct nearest-neighbor interaction between the immobile particles, the both systems can be driven from a phase-separated state (the segregated phase) to a crystalline state (the chessboard phase) and these transitions occur necessarily via charge-stripe phases: via a diagonal striped phase in the case of fermions and via vertical (horizontal) striped phases in the case of hardcore bosons. Secondly, the phase diagrams of the two systems (mobile fermions or mobile hardcore bo