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Abstract: Abstract We consider level-set percolation for the Gaussian membrane model on \(\mathbb Z^d\) , with \(d \ge 5\) , and establish that as \(h \in {\mathbb {R}}\) varies, a non-trivial percolation phase transition for the level-set above level h occurs at some finite critical level \(h_*\) , which we show to be positive in high dimensions. Along \(h_*\) , two further natural critical levels \(h_{**}\) and \({\overline{h}}\) are introduced, and we establish that \( -\infty<{\overline{h}} \le h_*\le h_{**} < \infty \) , in all dimensions. For \(h > h_{**}\) , we find that the connectivity function of the level-set above h admits stretched exponential decay, whereas for \(h < {\overline{h}}\) , chemical distances in the (unique) infinite cluster of the level-set are shown to be comparable to the Euclidean distance, by verifying conditions identified by Drewitz et al. (J Math Phys 55(8):083307, 2014) for general correlated percolation models. As a pivotal tool to study its level-set, we prove novel decoupling inequalities for the membrane model. PubDate: 2023-01-24

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Abstract: Abstract We consider the zeros of the partition function of the Ising model with ferromagnetic pair interactions and complex external field. Under the assumption that the graph with strictly positive interactions is connected, we vary the interaction (denoted by t) at a fixed edge. It is already known that each zero is monotonic (either increasing or decreasing) in t; we prove that its motion is local: the entire trajectories of any two distinct zeros are disjoint. If the underlying graph is a complete graph and all interactions take the same value \(t\ge 0\) (i.e., the Curie-Weiss model), we prove that all the principal zeros (those in \(i[0,\pi /2)\) ) decrease strictly in t. PubDate: 2023-01-19

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Abstract: Abstract We consider bootstrap percolation and diffusion in sparse random graphs with fixed degrees, constructed by configuration model. Every vertex has two states: it is either active or inactive. We assume that to each vertex is assigned a nonnegative (integer) threshold. The diffusion process is initiated by a subset of vertices with threshold zero which consists of initially activated vertices, whereas every other vertex is inactive. Subsequently, in each round, if an inactive vertex with threshold \(\theta \) has at least \(\theta \) of its neighbours activated, then it also becomes active and remains so forever. This is repeated until no more vertices become activated. The main result of this paper provides a central limit theorem for the final size of activated vertices. Namely, under suitable assumptions on the degree and threshold distributions, we show that the final size of activated vertices has asymptotically Gaussian fluctuations. PubDate: 2023-01-19

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Abstract: Abstract In this paper, for supersymmetric quantum integrable spin chains with rational \(Gl(N M)\) -invariant \(R\) -matrices, we construct a coupled master \(T\) -operator which represents a generating function for two-folds commuting quantum transfer matrices. We show that the functional relations for the quantum transfer matrices are equivalent to an infinite set of Hirota bilinear equations of the modified universal character hierarchy. Also the free fermion representation of the tau function of the supersymmetric quantum two-component spin chains will be given with the help of two sets of Clifford algebras. PubDate: 2023-01-18

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Abstract: Abstract We obtain a quenched vector-valued almost sure invariance principle (ASIP) for random expanding on average cocycles. This is achieved by combining the adapted version of Gouëzel’s approach for establishing ASIP (developed in Dragičević and Hafouta in Nonlinearity 34:6773–6798, 2021) and the recent construction of the so-called adapted norms (carried out in Dragičević and Sedro in Quenched limit theorems for expanding on average cocycles. arXiv:2105.00548, 2021), which in some sense eliminate the non-uniformity of the decay of correlations. For real-valued observables, we also show that the martingale approximation technique is applicable in our setup, and that it yields the ASIP with better error rates. Finally, we present an example showing the necessity of the scaling condition (12), answering a question of Dragičević and Sedro in Quenched limit theorems for expanding on average cocycles. arXiv:2105.00548, 2021. PubDate: 2023-01-18

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Abstract: Abstract The coupling through both drag force and volume fraction (of gas) of a kinetic equation of Vlasov type and a system of Euler or Navier–Stokes type (in which the volume fraction explicity appears) leads to the so-called thick sprays equations. Those equations are used to describe sprays (droplets or dust specks in a surrounding gas) in which the volume fraction of the disperse phase is non negligible. As for other multiphase flows systems, the issues related to the linear stability around homogeneous solutions is important for the applications. We show in this paper that this stability indeed holds for thick sprays equations, under physically reasonable assumptions. The analysis which is performed makes use of Lyapunov functionals for the linearized equations. PubDate: 2023-01-11

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Abstract: Abstract We prove the global existence of the non-negative unique mild solution for the Cauchy problem of the cutoff Boltzmann equation for soft potential model \(-1\le \gamma < 0\) with the small initial data in three dimensional space. Thus our result fixes the gap for the case \(\gamma =-1\) in three dimensional space in the authors’ previous work (He and Jiang in J Stat Phys 168(2):470–481, 2017) where the estimate for the loss term was improperly used. The other gap in He and Jiang (2017) for the case \(\gamma =0\) in two dimensional space is recently fixed by Chen et al. (Arch Ration Mech Anal 240:327–381, 2021). The initial data \(f_{0}\) is non-negative and satisfies that \(\Vert \langle v \rangle ^{\ell _{\gamma }} f_{0}(x,v)\Vert _{L^{3}_{x,v}}\ll 1\) and \(\Vert \langle v \rangle ^{\ell _{\gamma }} f_0\Vert _{L^{15/8}_{x,v}}<\infty \) where \(\ell _{\gamma }=0\) when \(\gamma =-1\) and \(\ell _{\gamma }=(1+\gamma )^{+}\) when \(-1<\gamma <0\) . We also show that the solution scatters with respect to the kinetic transport operator. The novel contribution of this work lies in the exploration of the symmetric property of the gain term in terms of weighted estimate. It is the key ingredient for solving the model \(-1<\gamma <0\) when applying the Strichartz estimates. PubDate: 2023-01-11

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Abstract: Abstract Regression models usually tend to recover a noisy signal in the form of a combination of regressors, also called features in machine learning, themselves being the result of a learning process. The alignment of the prior covariance feature matrix with the signal is known to play a key role in the generalization properties of the model, i.e. its ability to make predictions on unseen data during training. We present a statistical physics picture of the learning process. First we revisit the ridge regression to obtain compact asymptotic expressions for train and test errors, rendering manifest the conditions under which efficient generalization occurs. It is established thanks to an exact test-train sample error ratio combined with random matrix properties. Along the way in the form of a self-energy emerges an effective ridge penalty—precisely the train to test error ratio—which offers a very simple parameterization of the problem. This formulation appears convenient to tackle the learning process of the feature matrix itself. We derive an autonomous dynamical system in terms of elementary degrees of freedom of the problem determining the evolution of the relative alignment between the population matrix and the signal. A macroscopic counterpart of these equations is also obtained and various dynamical mechanisms are unveiled, allowing one to interpret the dynamics of simulated learning processes and to reproduce trajectories of single experimental run with high precision. PubDate: 2023-01-10

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Abstract: Abstract Sharp large deviation estimates for stochastic differential equations with small noise, based on minimizing the Freidlin–Wentzell action functional under appropriate boundary conditions, can be obtained by integrating certain matrix Riccati differential equations along the large deviation minimizers or instantons, either forward or backward in time. Previous works in this direction often rely on the existence of isolated minimizers with positive definite second variation. By adopting techniques from field theory and explicitly evaluating the large deviation prefactors as functional determinant ratios using Forman’s theorem, we extend the approach to general systems where degenerate submanifolds of minimizers exist. The key technique for this is a boundary-type regularization of the second variation operator. This extension is particularly relevant if the system possesses continuous symmetries that are broken by the instantons. We find that removing the vanishing eigenvalues associated with the zero modes is possible within the Riccati formulation and amounts to modifying the initial or final conditions and evaluation of the Riccati matrices. We apply our results in multiple examples including a dynamical phase transition for the average surface height in short-time large deviations of the one-dimensional Kardar–Parisi–Zhang equation with flat initial profile. PubDate: 2023-01-09

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Abstract: Abstract We derive macroscopic equations for a generalized contact process that is inspired by a neuronal integrate and fire model on the lattice \(\mathbb {Z}^d\) . The states at each lattice site can take values in \(0,\ldots ,k\) . These can be interpreted as neuronal membrane potential, with the state k corresponding to a firing threshold. In the terminology of the contact processes, which we shall use in this paper, the state k corresponds to the individual being infectious (all other states are noninfectious). In order to reach the firing threshold, or to become infectious, the site must progress sequentially from 0 to k. The rate at which it climbs is determined by other neurons at state k, coupled to it through a Kac-type potential, of range \(\gamma ^{-1}\) . The hydrodynamic equations are obtained in the limit \(\gamma \rightarrow 0\) . Extensions of the microscopic model to include excitatory and inhibitory neuron types, as well as other biophysical mechanisms, are also considered. PubDate: 2023-01-09

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Abstract: Abstract The paper is devoted to study of radially symmetric solutions to the spatially homogeneous Landau kinetic equation for Coulomb forces and related model equations. Two kinds of kinetic models for the Landau equation are introduced in order to understand a possibility of blow-up of solution. They differ by a diffusion coefficient. The model of the first kind is exactly solvable. Its global in time solution is constructed and studied. The model of the second type is more complicated. The power moments (in velocities) for this model and for the Landau equations are studied. The propagation in time of the exponential moment of the third order is proved for solutions of the Landau equation. Informally speaking, this means that a typical high-energy tail for solutions having indata with compact support looks like \( \exp [- b(t) v ^{k}] \) with some \( k \ge 3 \) . In particular, this means that a lower bound with Maxwellian tail \( \exp [- a(t) v ^{2}] \) is impossible for solutions of the Landau equation PubDate: 2023-01-07

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Abstract: Abstract In this paper we analyze the metastable behavior for the Ising model that evolves under Kawasaki dynamics on the hexagonal lattice \({\mathbb {H}}^2\) in the limit of vanishing temperature. Let \(\varLambda \subset {\mathbb {H}}^2\) a finite set which we assume to be arbitrarily large. Particles perform simple exclusion on \(\varLambda \) , but when they occupy neighboring sites they feel a binding energy \(-U<0\) . Along each bond touching the boundary of \(\varLambda \) from the outside to the inside, particles are created with rate \(\rho =e^{-\varDelta \beta }\) , while along each bond from the inside to the outside, particles are annihilated with rate 1, where \(\beta \) is the inverse temperature and \(\varDelta >0\) is an activity parameter. For the choice \(\varDelta \in {(U,\frac{3}{2}U)}\) we prove that the empty (resp. full) hexagon is the unique metastable (resp. stable) state. We determine the asymptotic properties of the transition time from the metastable to the stable state and we give a description of the critical configurations. We show how not only their size but also their shape varies depending on the thermodynamical parameters. Moreover, we emphasize the role that the specific lattice plays in the analysis of the metastable Kawasaki dynamics by comparing the different behavior of this system with the corresponding system on the square lattice. PubDate: 2023-01-06

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Abstract: Abstract We consider a two-species simple exclusion process on a periodic lattice. We use the method of matched asymptotics to derive evolution equations for the two population densities in the dilute regime, namely a cross-diffusion system of partial differential equations for the two species’ densities. First, our result captures non-trivial interaction terms neglected in the mean-field approach, including a non-diagonal mobility matrix with explicit density dependence. Second, it generalises the rigorous hydrodynamic limit of Quastel (Commun Pure Appl Math 45(6):623–679, 1992), valid for species with equal jump rates and given in terms of a non-explicit self-diffusion coefficient, to the case of unequal rates in the dilute regime. In the equal-rates case, by combining matched asymptotic approximations in the low- and high-density limits, we obtain a cubic polynomial approximation of the self-diffusion coefficient that is numerically accurate for all densities. This cubic approximation agrees extremely well with numerical simulations. It also coincides with the Taylor expansion up to the second-order in the density of the self-diffusion coefficient obtained using a rigorous recursive method. PubDate: 2023-01-06

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Abstract: Abstract We derive a continuum mean-curvature flow as a certain hydrodynamic scaling limit of Glauber-Kawasaki dynamics with speed change. The Kawasaki part describes the movement of particles through particle interactions. It is speeded up in a diffusive space-time scaling. The Glauber part governs the creation and annihilation of particles. The Glauber part is set to favor two levels of particle density. It is also speeded up in time, but at a lesser rate than the Kawasaki part. Under this scaling, a mean-curvature interface flow emerges, with a homogenized ‘surface tension-mobility’ parameter reflecting microscopic rates. The interface separates the two levels of particle density. Similar hydrodynamic limits have been derived in two recent papers; one where the Kawasaki part describes simple nearest neighbor interactions, and one where the Kawasaki part is replaced by a zero-range process. We extend the main results of these two papers beyond nearest-neighbor interactions. The main novelty of our proof is the derivation of a ‘Boltzmann-Gibbs’ principle which covers a class of local particle interactions. PubDate: 2023-01-02

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Abstract: Abstract We calculate the susceptibility of a canonical ensemble of quantum oscillators to the singular random metric. If the covariance of the metric is \(\vert \textbf{x}-\textbf{x}^{\prime }\vert ^{-4\alpha }\) ( \(0<\alpha <\frac{1}{2}\) )then the expansion of the partition function in powers of the temperature involves non-integer indices. PubDate: 2023-01-02

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Abstract: Abstract In this paper, we study phase transitions of a class of time-inhomogeneous diffusion processes associated with the \(\varphi ^4\) model. We prove that when \(\gamma <0\) , the system has no phase transition and when \(\gamma >0\) , the system has a phase transition and we study the phase transition of the system through qualitative and quantitative methods. We further show that, as the strength of the mean field tends to 0, the solution and stationary distribution of such system converge locally uniformly in \(L^2\) and Wasserstein distance respectively to those of corresponding system without mean field. PubDate: 2023-01-02

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Abstract: Abstract We study a stochastic process which describes the dynamics of a particle performing a finite-velocity random motion whose velocities alternate cyclically. We consider two cases, in which the state-space of the process is (i) \({\mathbb {R}} \times \{\vec {v}_1, \vec {v}_2\}\) , with velocities \(v_1 > v_2\) , and (ii) \({\mathbb {R}}^2 \times \{\vec {v}_1, \vec {v}_2, \vec {v}_3\}\) , where the particle moves along three different directions with possibly unequal velocities. Assuming that the random intertimes between consecutive changes of directions are governed by geometric counting processes, we first construct the stochastic models of the particle motion. Then, we investigate various features of the considered processes and obtain the formal expression of their probability laws. In the case (ii) we study a planar random motion with three specific directions and determine the exact transition probability density functions of the process when the initial velocity is fixed. In both cases we also investigate the behavior of the probability distributions of the motion when the intensities of the underlying geometric counting processes tend to infinity. The asymptotic probability law of the particle is found to be a uniform distribution in case (i) and a three-peaked distribution in case (ii). PubDate: 2023-01-02

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Abstract: Abstract In this paper we present a numerical study of a mathematical model of spiking neurons introduced by Ferrari et al. (J Stat Phys 172(6):1564–1575, 2018). In this model we have a countable number of neurons linked together in a network, each of them having a membrane potential taking value in the integers, and each of them spiking over time at a rate which depends on the membrane potential through some rate function \(\phi \) . Beside being affected by a spike each neuron can also be affected by leaking. At each of these leak times, which occur for a given neuron at a fixed rate \(\gamma \) , the membrane potential of the neuron concerned is spontaneously reset to 0. A wide variety of versions of this model can be considered by choosing different graph structures for the network and different activation functions. It was rigorously shown that when the graph structure of the network is the one-dimensional lattice with a hard threshold for the activation function, this model presents a phase transition with respect to \(\gamma \) , and that it also presents a metastable behavior. By the latter we mean that in the sub-critical regime the re-normalized time of extinction converges to an exponential random variable of mean 1. It has also been proven that in the super-critical regime the renormalized time of extinction converges in probability to 1. Here, we investigate numerically a richer class of graph structures and activation functions. Namely we investigate the case of the two dimensional and the three dimensional lattices, as well as the case of a linear function and a sigmoid function for the activation function. We present numerical evidence that the result of metastability in the sub-critical regime holds for these graphs and activation functions as well as the convergence in probability to 1 in the super-critical regime. PubDate: 2022-12-29

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Abstract: Abstract We consider simple models where particles interact with thermal traps: the particles move freely, except when they hit the traps where there are exchanges of mass, momentum and energy. We investigate the relaxation properties of the model, and we show that indeed the solutions relax towards a uniform Maxwellian state, entirely determined by the initial data. PubDate: 2022-12-28