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- Long-Range First-Passage Percolation on the Torus
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Abstract: Abstract We study a geometric version of first-passage percolation on the complete graph, known as long-range first-passage percolation. Here, the vertices of the complete graph \(\mathcal {K}_n\) are embedded in the d-dimensional torus \(\mathbb T_n^d\) , and each edge e is assigned an independent transmission time \(T_e=\Vert e\Vert _{\mathbb T_n^d}^\alpha E_e\) , where \(E_e\) is a rate-one exponential random variable associated with the edge e, \(\Vert \cdot \Vert _{\mathbb T_n^d}\) denotes the torus-norm, and \(\alpha \ge 0\) is a parameter. We are interested in the case \(\alpha \in [0,d)\) , which corresponds to the instantaneous percolation regime for long-range first-passage percolation on \(\mathbb {Z}^d\) studied by Chatterjee and Dey [14], and which extends first-passage percolation on the complete graph (the \(\alpha =0\) case) studied by Janson [24]. We consider the typical distance, flooding time, and diameter of the model. Our results show a 1, 2, 3-type result, akin to first-passage percolation on the complete graph as shown by Janson. The results also provide a quantitative perspective to the qualitative results observed by Chatterjee and Dey on \(\mathbb {Z}^d\) .
PubDate: 2024-08-27
- Faithfulness of Real-Space Renormalization Group Maps
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Abstract: Abstract The behavior of \(b=2\) real-space renormalization group (RSRG) maps like the majority rule and the decimation map was examined by numerically applying RSRG steps to critical \(q=2,3,4\) Potts spin configurations. While the majority rule is generally believed to work well, a more thorough investigation of the action of the map has yet to be considered in the literature. When fixing the size of the renormalized lattice \(L_g\) and allowing the source configuration size \(L_0\) to vary, we observed that the RG flow of the spin and energy correlation under the majority rule map appear to converge to a nontrivial model-dependent curve. We denote this property as “faithfulness”, because it implies that some information remains preserved by RSRG maps that fall under this class. Furthermore, we show that \(b=2\) weighted majority-like RSRG maps acting on the \(q=2\) Potts model can be divided into two categories, maps that behave like decimation and maps that behave like the majority rule.
PubDate: 2024-08-27
- Infinite-Volume Gibbs States of the Generalized Mean-Field Orthoplicial
Model-
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Abstract: Abstract The generalized mean-field orthoplicial model is a mean-field model on a space of continuous spins on \(\mathbb {R}^n\) that are constrained to a scaled \((n-1)\) -dimensional \(\ell _1\) -sphere, equivalently a scaled \((n-1)\) -dimensional orthoplex, and interact through a general interaction function. The finite-volume Gibbs states of this model correspond to singular probability measures. In this paper, we use probabilistic methods to rigorously classify the infinite-volume Gibbs states of this model, and we show that they are convex combinations of product states. The predominant methods utilize the theory of large deviations, relative entropy, and equivalence of ensembles, and the key technical tools utilize exact integral representations of certain partition functions and locally uniform estimates of expectations of certain local observables.
PubDate: 2024-08-27
- Cutoff Ergodicity Bounds in Wasserstein Distance for a Viscous Energy
Shell Model with Lévy Noise-
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Abstract: Abstract This article establishes explicit non-asymptotic ergodic bounds in the renormalized Wasserstein–Kantorovich–Rubinstein (WKR) distance for a viscous energy shell lattice model of turbulence with random energy injection. The system under consideration is driven either by a Brownian motion, a symmetric \(\alpha \) -stable Lévy process, a stationary Gaussian or \(\alpha \) -stable Ornstein–Uhlenbeck process, or by a general Lévy process with second moments. The obtained non-asymptotic bounds establish asymptotically abrupt thermalization. The analysis is based on the explicit representation of the solution of the system in terms of convolutions of Bessel functions.
PubDate: 2024-08-27
- Functional Limit Theorems for Dynamical Systems with Correlated Maximal
Sets-
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Abstract: Abstract In order to obtain functional limit theorems for heavy-tailed stationary processes arising from dynamical systems, one needs to understand the clustering patterns of the tail observations of the process. These patterns are well described by means of a structure called the pilling process introduced recently in the context of dynamical systems. So far, the pilling process has been computed only for observable functions maximised at a single repelling fixed point. Here, we study richer clustering behaviours by considering correlated maximal sets, in the sense that the observable is maximised in multiple points belonging to the same orbit, and we work out explicit expressions for the pilling process when the dynamics is piecewise linear and expanding (1-dimensional and 2-dimensional).
PubDate: 2024-08-27
- Inferring Kinetics and Entropy Production from Observable Transitions in
Partially Accessible, Periodically Driven Markov Networks-
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Abstract: Abstract For a network of discrete states with a periodically driven Markovian dynamics, we develop an inference scheme for an external observer who has access to some transitions. Based on waiting-time distributions between these transitions, the periodic probabilities of states connected by these observed transitions and their time-dependent transition rates can be inferred. Moreover, the smallest number of hidden transitions between accessible ones and some of their transition rates can be extracted. We prove and conjecture lower bounds on the total entropy production for such periodic stationary states. Even though our techniques are based on generalizations of known methods for steady states, we obtain original results for those as well.
PubDate: 2024-08-14
- Diameters of Symmetric and Lifted Simple Exclusion Models
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Abstract: Abstract We determine diameters of Markov chains describing one-dimensional N-particle models with an exclusion interaction, namely the symmetric simple exclusion process (Ssep) and one of its non-reversible liftings, the lifted totally asymmetric simple exclusion process (Tasep). The diameters provide lower bounds for the mixing times, and we discuss the implications of our findings for the analysis of these models.
PubDate: 2024-08-13
- Correction to: Entropy of Irregular Points for Some Dynamical Systems
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PubDate: 2024-08-12
- Existence and Uniqueness of Nonsimple Multiple SLE
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Abstract: Abstract We prove the existence and uniqueness of multiple \(\hbox {SLE}_{\kappa }\) associated with any given link pattern for \(\kappa \in (4,6]\) . We also have the uniqueness for \(\kappa \in (6,8)\) . The multiple \(\hbox {SLE}_{\kappa }\) law is constructed by first inductively constructing a \(\sigma \) -finite multiple \(\hbox {SLE}_{\kappa }\) measure and then normalizing the measure whenever it is finite. The total mass of the measure satisfies the conformal covariance, asymptotics and PDE for multiple \(\hbox {SLE}_{\kappa }\) partition functions in the literature subject to the assumption that it is smooth.
PubDate: 2024-08-11
- Prethermalization and Conservation Laws in Quasi-Periodically Driven
Quantum Systems-
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Abstract: Abstract We study conservation laws of a general class of quantum many-body systems subjected to an external time dependent quasi-periodic driving. When the frequency of the driving is large enough or the strength of the driving is small enough, we prove a Nekhoroshev-type stability result: we show that the system exhibits a prethermal state for stretched exponentially long times in the perturbative parameter. Moreover, we prove the quasi-conservation of the constants of motion of the unperturbed Hamiltonian and we analyze their physical meaning in examples of relevance to condensed matter and statistical physics.
PubDate: 2024-08-10
- Random Normal Matrices: Eigenvalue Correlations Near a Hard Wall
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Abstract: Abstract We study pair correlation functions for planar Coulomb systems in the pushed phase, near a ring-shaped impenetrable wall. We assume coupling constant \(\Gamma =2\) and that the number n of particles is large. We find that the correlation functions decay slowly along the edges of the wall, in a narrow interface stretching a distance of order 1/n from the hard edge. At distances much larger than \(1/\sqrt{n}\) , the effect of the hard wall is negligible and pair correlation functions decay very quickly, and in between sits an interpolating interface that we call the “semi-hard edge”. More precisely, we provide asymptotics for the correlation kernel \(K_{n}(z,w)\) as \(n\rightarrow \infty \) in two microscopic regimes (with either \( z-w = \mathcal{O}(1/\sqrt{n})\) or \( z-w = \mathcal{O}(1/n)\) ), as well as in three macroscopic regimes (with \( z-w \asymp 1\) ). For some of these regimes, the asymptotics involve oscillatory theta functions and weighted Szegő kernels.
PubDate: 2024-08-08
- Spin Glass to Paramagnetic Transition and Triple Point in Spherical SK
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Abstract: Abstract This paper studies spin glass to paramagnetic transition in the Spherical Sherrington–Kirkpatrick model with ferromagnetic Curie-Weiss interaction with coupling constant J and inverse temperature \(\beta \) . The disorder of the system is represented by a general Wigner matrix. We confirm a conjecture of Baik and Lee (Stat Phys 165(2):185–224, 2016; Ann Henri Poincaré 18(6):1867–1917, 2017), that the critical window of temperatures for this transition is \(\beta = 1 + bN^{-1/3} \sqrt{\log N}\) with \(b\in \mathbb {R}\) . The limiting distribution of the scaled free energy is Gaussian for negative b and a weighted linear combination of independent Gaussian and Tracy–Widom components for positive b. In the special case where the Wigner matrix is from the Gaussian Orthogonal or Unitary Ensemble, we describe the triple point transition between spin glass, paramagnetic, and ferromagnetic regimes in a critical window for \((\beta , J)\) around the triple point (1, 1): the Tracy–Widom component is replaced by the one parameter family of deformations described by Bloemendal and Virág [9].
PubDate: 2024-08-08
- Phase Transition of Eigenvalues in Deformed Ginibre Ensembles II: GinSE
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Abstract: Abstract Consider a random matrix of size 2N as an additive deformation of the quaternionic Ginibre ensemble under a deterministic matrix \(X_0\) with a finite rank, independent of N. When some eigenvalues of \(X_0\) lie inside the unit disk, we prove that local eigenvalue statistics at the real edge is governed by a new class of Pfaffian point processes, for which correlation kernels only depend on geometric multiplicity of eigenvalue, while at the complex edge it is governed by the same statistics as recently found in the deformed GinUE.
PubDate: 2024-08-08
- Large Deviations of Invariant Measures of Stochastic Reaction–Diffusion
Equations on Unbounded Domains-
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Abstract: Abstract This paper is concerned with the large deviation principle of invariant measures of the stochastic reaction–diffusion equation with polynomial drift driven by additive noise defined on the entire space \(\mathbb {R}^n\) . Since the standard Sobolev embeddings on \(\mathbb {R}^n\) are not compact and the spectrum of the Laplace operator on \(\mathbb {R}^n\) are not discrete, there are many issues for proving the large deviations of invariant measures in the case of unbounded domains, including the difficulties for proving the compactness of the level sets of rate functions, the uniform Dembo–Zeitouni large deviations on compact sets as well as the exponential tightness on compact sets. Currently, there is no result available in the literature on the large deviations of invariant measures for stochastic PDEs on unbounded domains, and this paper is the first one to deal with this issue. The non-compactness of the standard Sobolev embeddings on \(\mathbb {R}^n\) is circumvented by the idea of uniform tail-ends estimates together with the arguments of weighted spaces.
PubDate: 2024-08-03
- A Novel ES-BGK Model for Non-polytropic Gases with Internal State Density
Independent of the Temperature-
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Abstract: Abstract A novel ES-BGK-based model of non-polytropic rarefied gases in the framework of kinetic theory is presented. Key features of this model are: an internal state density function depending only on the microscopic energy of internal modes (avoiding the dependence on temperature seen in previous reference studies); full compliance with the H-theorem; feasibility of the closure of the system of moment equations based on the maximum entropy principle, following the well-established procedure of rational extended thermodynamics. The structure of planar shock waves in carbon dioxide (CO \(_2\) ) obtained with the present model is in general good agreement with that of previous results, except for the computed internal temperature profile, which is qualitatively different with respect to the results obtained in previous studies, showing here a consistently monotonic behavior across the shock structure, rather than the non monotonic behavior previously found.
PubDate: 2024-07-30
- Maximum of the Gaussian Interface Model in Random External Fields
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Abstract: Abstract We consider the Gaussian interface model in the presence of random external fields, that is the finite volume (random) Gibbs measure on \(\mathbb {R}^{\Lambda _N}\) , \(\Lambda _N=[-N, N]^d\cap \mathbb {Z}^d\) with Hamiltonian \(H_N(\phi )= \frac{1}{4d}\sum \limits _{x\sim y}(\phi (x)-\phi (y))^2 -\sum \limits _{x\in \Lambda _N}\eta (x)\phi (x)\) and 0-boundary conditions. \(\{\eta (x)\}_{x\in \mathbb {Z}^d}\) is a family of i.i.d. symmetric random variables. We study how the typical maximal height of a random interface is modified by the addition of quenched bulk disorder. We show that the asymptotic behavior of the maximum changes depending on the tail behavior of the random variable \(\eta (x)\) when \(d\ge 5\) . In particular, we identify the leading order asymptotics of the maximum.
PubDate: 2024-07-27
- Time Evolution of the Boltzmann Entropy for a Nonequilibrium Dilute Gas
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Abstract: Abstract We investigate the time evolution of the Boltzmann entropy of a dilute gas of N particles, \(N\gg 1\) , as it undergoes a free expansion doubling its volume. The microstate of the system changes in time via Hamiltonian dynamics. Its entropy, at any time t, is given by the logarithm of the phase space volume of all the microstates giving rise to its macrostate at time t. The macrostates that we consider are defined by coarse graining the one-particle phase space into cells \(\Delta _\alpha \) . The initial and final macrostates of the system are thermal equilibrium states in volumes V and 2V, with the same energy E and particle number N. Their entropy per particle is given, for sufficiently large systems, by the thermodynamic entropy as a function of the particle and energy density, whose leading term is independent of the size of the \(\Delta _\alpha \) . The intermediate (non-equilibrium) entropy does however depend on the size of the cells \(\Delta _\alpha \) . Its change with time is due to (i) dispersal in physical space from free motion and to (ii) the collisions between particles which change their velocities. The former depends strongly on the size of the velocity coarse graining \(\Delta v\) : it produces entropy at a rate proportional to \(\Delta v\) . This dependence is investigated numerically and analytically for a dilute two-dimensional gas of hard discs. It becomes significant when the mean free path between collisions is of the same order or larger than the length scale of the initial spatial inhomogeneity. In the opposite limit, the rate of entropy production is essentially independent of \(\Delta v\) and is given by the Boltzmann equation for the limit \(\Delta v\rightarrow 0\) . We show that when both processes are active the time dependence of the entropy has a scaling form involving the ratio of the rates of its production by the two processes.
PubDate: 2024-07-26
- Maximal Codimension Collisions and Invariant Measures for Hard Spheres on
a Line-
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Abstract: Abstract For any \(N\ge 3\) , we study invariant measures of the dynamics of N hard spheres whose centres are constrained to lie on a line. In particular, we study the invariant submanifold \(\mathcal {M}\) of the tangent bundle of the hard sphere billiard table comprising initial data that lead to the simultaneous collision of all N hard spheres. Firstly, we obtain a characterisation of those continuously-differentiable N-body scattering maps which generate a billiard dynamics on \(\mathcal {M}\) admitting a canonical weighted Hausdorff measure on \(\mathcal {M}\) (that we term the Liouville measure on \(\mathcal {M}\) ) as an invariant measure. We do this by deriving a second boundary-value problem for a fully nonlinear PDE that all such scattering maps satisfy by necessity. Secondly, by solving a family of functional equations, we find sufficient conditions on measures which are absolutely continuous with respect to the Hausdorff measure in order that they be invariant for billiard flows that conserve momentum and energy. Finally, we show that the unique momentum- and energy-conserving linear N-body scattering map yields a billiard dynamics which admits the Liouville measure on \(\mathcal {M}\) as an invariant measure.
PubDate: 2024-07-26
- Gelation and Localization in Multicomponent Coagulation with
Multiplicative Kernel Through Branching Processes-
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Abstract: Abstract The multicomponent coagulation equation is a generalization of the Smoluchowski coagulation equation, where the size of a particle is described by a vector. Similar to the original Smoluchowski equation, the multicomponent coagulation equation exhibits gelation behavior when supplied with a multiplicative kernel. Additionally, a new type of behaviour called localization is observed due to the multivariate nature of the particle size distribution. Here we extend the branching process representation technique, which we introduced to study differential equations in our previous work, and apply it to find a concise probabilistic solution of the multicomponent coagulation equation supplied with monodisperse initial conditions. We also provide short proofs for the gelation time and characterisation the localization phenomenon.
PubDate: 2024-07-23
- A New Model for Preferential Attachment Scheme with Time-Varying
Parameters-
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Abstract: Abstract We propose an extension of the preferential attachment scheme by allowing the connecting probability to depend on time t. We estimate the parameters involved in the model by minimizing the expected squared difference between the number of vertices of degree one and its conditional expectation. The asymptotic properties of the estimators are also investigated when the parameters are time-varying by establishing the central limit theorem (CLT) of the number of vertices of degree one. We propose a new statistic to test whether the parameters have change points. We also offer some methods to estimate the number of change points and detect the locations of change points. Simulations are conducted to illustrate the performances of the above results.
PubDate: 2024-07-19
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