Journal of Statistical Physics
Journal Prestige (SJR): 0.93 Citation Impact (citeScore): 1 Number of Followers: 12 Hybrid journal (It can contain Open Access articles) ISSN (Print) 15729613  ISSN (Online) 00224715 Published by SpringerVerlag [2469 journals] 
 Exact Computation of GrowthRate Variance in Randomly Fluctuating
Environment
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Abstract: Abstract We consider a general class of Markovian models describing the growth in a randomly fluctuating environment of a clonal biological population having several phenotypes related by stochastic switching. Phenotypes differ e.g. by the level of gene expression for a population of bacteria. The timeaveraged growth rate of the population, \(\Lambda \) , is selfaveraging in the limit of infinite time; it may be understood as the fitness of the population in a context of Darwinian evolution. The observation time T being however typically finite, the growth rate fluctuates. For T finite but large, we obtain the variance of the timeaveraged growth rate as the maximum of a functional based on the stationary probability distribution for the phenotypes. This formula is general. In the case of two states, the stationary probability was computed by Hufton et al. (J Stat Mech 2018:23501, 2018), allowing for an explicit expression which can be checked numerically. Applications of our main formula to the study of survival strategies of biological populations, as developed in the companion article (Dinis et al. in http://doi.org/10.1101/2022.01.18.476793v1), are presented here briefly.
PubDate: 20221003

 Perturbation of Systems with a First Integral: Motion on the Reeb Graph

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Abstract: Abstract We consider the longtime behavior of systems close to a system with a smooth first integral. Under certain assumptions, the limiting behavior, to some extent, turns out to be universal: it is determined by the first integral, the deterministic perturbation, and the initial point. Furthermore, it is the same for a broad class of noises. In particular, the longtime behavior of a deterministic system can, in a sense, be stochastic and stochastic systems can have a reduced stochasticity. The limiting distribution is calculated explicitly.
PubDate: 20220929

 Local Central Limit Theorem for LongRange TwoBody Potentials at
Sufficiently High Temperatures
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Abstract: Abstract Dobrushin and Tirozzi (Commun Math Phys 54(2):173–192, 1977) showed that, for a Gibbs measure with the finiterange potential, the Local Central Limit Theorem is implied by the Integral Central Limit Theorem. Campanino et al. (Commun Math Phys 70(2):125–132, 1979) extended this result for a family of Gibbs measures for longrange pair potentials satisfying certain conditions. We are able to show for a family of Gibbs measures for longrange pair potentials not satisfying the conditions given in Campanino et al. (Commun Math Phys 70(2):125–132, 1979) , that at sufficiently high temperatures, if the Integral Central Limit Theorem holds for a given sequence of Gibbs measures, then the Local Central Limit Theorem also holds for the same sequence. We also extend (Campanino et al. in Commun Math Phys 70(2):125–132, 1979) to the case when the state space is general, provided that it is equipped with a finite measure.
PubDate: 20220926

 Joint Invariance Principles for Random Walks with Positively and
Negatively Reinforced Steps
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Abstract: Abstract Given a random walk \((S_n)\) with typical step distributed according to some fixed law and a fixed parameter \(p \in (0,1)\) , the associated positively stepreinforced random walk is a discretetime process which performs at each step, with probability \(1p\) , the same step as \((S_n)\) while with probability p, it repeats one of the steps it performed previously chosen uniformly at random. The negatively stepreinforced random walk follows the same dynamics but when a step is repeated its sign is also changed. In this work, we shall prove functional limit theorems for the triplet of a random walk, coupled with its positive and negative reinforced versions when \(p < 1/2\) and when the typical step is centred. The limiting process is Gaussian and admits a simple representation in terms of stochastic integrals, $$\begin{aligned} \left( B(t) , \, t^p \int _0^t s^{p} \mathrm {d}B^r(s) , \, t^{p} \int _0^t s^{p} \mathrm {d}B^c(s) \right) _{t \in \mathbb {R}^+} \end{aligned}$$ for properly correlated Brownian motions \(B, B^r\) , \(B^c\) . The processes in the second and third coordinate are called the noise reinforced Brownian motion (as named in [1]), and the noise counterbalanced Brownian motion of B. Different couplings are also considered, allowing us in some cases to drop the centredness hypothesis and to completely identify for all regimes \(p \in (0,1)\) the limiting behaviour of step reinforced random walks. Our method exhausts a martingale approach in conjunction with the martingale functional CLT.
PubDate: 20220926

 Cluster Expansions: Necessary and Sufficient Convergence Conditions

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Abstract: We prove a new convergence condition for the activity expansion of correlation functions in equilibrium statistical mechanics with possibly negative pair potentials. For nonnegative pair potentials, the criterion is an if and only if condition. The condition is formulated with a signflipped Kirkwood–Salsburg operator and known conditions such as Kotecký–Preiss and Fernández–Procacci are easily recovered. In addition, we deduce new sufficient convergence conditions for hardcore systems in \(\mathbb {R}^d\) and \(\mathbb {Z}^d\) as well as for abstract polymer systems. The latter improves on the Fernández–Procacci criterion.
PubDate: 20220924

 Nonequilibrium Stationary Properties of the Boundary Driven ZeroRange
Process with Long Jumps
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Abstract: Abstract We consider the zerorange process with long jumps and in contact with infinitely extended reservoirs in its nonequilibrium stationary state. We derive the hydrostatic limit and the Fick’s law, which are a consequence of a static relationship between the exclusion process and the zerorange process. We also obtain the large deviation principle for the empirical density, i.e. we compute the nonequilibrium free energy.
PubDate: 20220923

 Quasistatic Decomposition and the Gibbs Factorial in Small Thermodynamic
Systems
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Abstract: Abstract For a classical system consisting of Ninteracting identical particles in contact with a heat bath, we define the free energy from thermodynamic relations in equilibrium statistical mechanics. Concretely, the temperature dependence of the free energy is determined from the GibbsHelmholtz relation, and its volume dependence is determined from the condition that the quasistatic work in a volume change is equal to the free energy change. Now, we argue the free energy difference in a quasistatic decomposition of small thermodynamic systems. We can then determine the N dependence of the free energy, which includes the Gibbs factorial N! in addition to the phase space integration of the Gibbs–Boltzmann factor.
PubDate: 20220921

 The Least Singular Value of the General Deformed Ginibre Ensemble

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Abstract: Abstract We study the least singular value of the \(n\times n\) matrix \(Hz\) with \(H=A_0+H_0\) , where \(H_0\) is drawn from the complex Ginibre ensemble of matrices with iid Gaussian entries, and \(A_0\) is some general \(n\times n\) matrix with complex entries (it can be random and in this case it is independent of \(H_0\) ). Assuming some rather general assumptions on \(A_0\) , we prove an optimal tail estimate on the least singular value in the regime where z is around the spectral edge of H thus generalize the recent result of Cipolloni et al. (Probab Math Phys 1(1):101–146, 2020) to the case \(A_0\ne 0\) . The result improves the classical bound by Sankar et al. (SIAM J Matrix Anal Appl 28:446–476, 2006).
PubDate: 20220920

 Convergence of the Free Energy for Spherical Spin Glasses

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Abstract: Abstract We prove that the free energy of any spherical mixed pspin model converges as the dimension N tends to infinity. While the convergence is a consequence of the Parisi formula, the proof we give is independent of the formula and uses the wellknown Guerra–Toninelli interpolation method. The latter was invented for models with Ising spins to prove that the free energy is superadditive and therefore (normalized by N) converges. In the spherical case, however, the configuration space is not a product space and the interpolation cannot be applied directly. We first relate the free energy on the sphere of dimension \(N+M\) to a free energy defined on the product of spheres in dimensions N and M to which we then apply the interpolation method. This yields an approximate superadditivity which is sufficient to prove the convergence.
PubDate: 20220918

 Phase Growth with Heat Diffusion in a Stochastic Lattice Model

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Abstract: Abstract When a stable phase is adjacent to a metastable phase with a planar interface, the stable phase grows. We propose a stochastic lattice model describing the phase growth accompanying heat diffusion. The model is based on an energyconserving Potts model with a kinetic energy term defined on a twodimensional lattice, where each site is sparserandomly connected in one direction and local in the other direction. For this model, we calculate the stable and metastable phases exactly using statistical mechanics. Performing numerical simulations, we measure the displacement of the interface R(t). We observe the scaling relation \(R(t)=L_x \bar{\mathcal {R}} (Dt/L_x^2)\) , where D is the thermal diffusion constant and \(L_x\) is the system size between the two heat baths. The scaling function \(\bar{\mathcal {R}}(z)\) shows \(\bar{\mathcal {R}}(z) \simeq z^{0.5}\) for \(z \ll z_c\) and \(\bar{\mathcal {R}}(z) \simeq z^{\alpha }\) for \(z \gg z_c\) , where the crossover value \(z_c\) and exponent \(\alpha \) depend on the temperatures of the baths, and \(0.5\le \alpha \le 1\) . We then confirm that a deterministic phasefield model exhibits the same scaling relation. Moreover, numerical simulations of the phasefield model show that the crossover value \(\bar{\mathcal {R}}(z_c)\) approaches zero when the stable phase becomes neutral.
PubDate: 20220918

 Algorithmic Pure States for the Negative Spherical Perceptron

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Abstract: Abstract We consider the spherical perceptron with Gaussian disorder. This is the set S of points \(\varvec{\sigma }\in \mathbb {R}^N\) on the sphere of radius \(\sqrt{N}\) satisfying \(\langle \varvec{g}_a , \varvec{\sigma }\rangle \ge \kappa \sqrt{N}\) for all \(1 \le a \le M\) , where \((\varvec{g}_a)_{a=1}^M\) are independent standard gaussian vectors and \(\kappa \in \mathbb {R}\) is fixed. Various characteristics of S such as its measure and the largest M for which it is nonempty, were computed heuristically in statistical physics in the asymptotic regime \(N \rightarrow \infty \) , \(M/N \rightarrow \alpha \) . The case \(\kappa <0\) is of special interest as S is conjectured to exhibit a hierarchical treelike geometry known as full replicasymmetry breaking ( \(\text {FRSB}\) ) close to the satisfiability threshold \(\alpha _{\text {SAT}}(\kappa )\) , whose characteristics are captured by a Parisi variational principle akin to the one appearing in the Sherrington–Kirkpatrick model. In this paper we design an efficient algorithm which, given oracle access to the solution of the Parisi variational principle, exploits this conjectured \(\text {FRSB}\) structure for \(\kappa <0\) and outputs a vector \(\hat{\varvec{\sigma }}\) satisfying \(\langle \varvec{g}_a , \hat{\varvec{\sigma }}\rangle \ge \kappa \sqrt{N}\) for all \(1\le a \le M\) and lying on a sphere of nontrivial radius \(\sqrt{\bar{q}N}\) , where \(\bar{q}\in (0,1)\) is the rightend of the support of the associated Parisi measure. We expect \(\hat{\varvec{\sigma }}\) to be approximately the barycenter of a pure state of the spherical perceptron. Moreover we expect that \(\bar{q}\rightarrow 1\) as \(\alpha \rightarrow \alpha _{\text {SAT}}(\kappa )\) , so that \(\big \langle \varvec{g}_a, \hat{\varvec{\sigma }} \big \rangle / \vert \hat{\varvec{\sigma }} \vert \ge \kappa o(1)\) near criticality.
PubDate: 20220916

 Displacement Correlations in Disordered Athermal Networks

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Abstract: Abstract We derive exact results for correlations in the displacement fields \(\{ \delta \mathbf {r} \} \equiv \{ \delta r_{\mu = x,y} \}\) in nearcrystalline athermal systems in two dimensions. We analyze the displacement correlations produced by different types of microscopic disorder, and show that disorder at the microscopic scale gives rise to longrange correlations with a dependence on the system size L given by \(\langle \delta r_{\mu } \delta r_{\nu } \rangle \sim c_{\mu \nu }(r/L,\theta )\) . In addition, we show that polydispersity in the constituent particle sizes and random bond disorder give rise to a logarithmic system size scaling, with \(c_{\mu \nu }(\rho ,\theta ) \sim \text {const}_{\mu \nu }  \text {a}_{\mu \nu }(\theta )\log \rho + \text {b}_{\mu \nu }(\theta ) \rho ^{2} \) for \(\rho ~(=r/L) \rightarrow 0\) . This scaling is different for the case of displacement correlations produced by random external forces at each vertex of the network, given by \(c^{f}_{\mu \nu }(\rho ,\theta ) \sim \text {const}^{f}_{\mu \nu } ( \text {a}^{f}_{\mu \nu }(\theta ) + \text {b}^{f}_{\mu \nu }(\theta ) \log \rho ) \rho ^2 \) . Additionally, we find that correlations produced by polydispersity and the correlations produced by disorder in bond stiffness differ in their symmetry properties. Finally, we also predict the displacement correlations for a model of polydispersed soft disks subject to external pinning forces, that involve two different types of microscopic disorder. We verify our theoretical predictions using numerical simulations of polydispersed soft disks with random spring contacts in two dimensions.
PubDate: 20220915

 Nonreversible Metastable Diffusions with Gibbs Invariant Measure II:
Markov Chain Convergence
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Abstract: Abstract This article considers a class of metastable nonreversible diffusion processes whose invariant measure is a Gibbs measure associated with a Morse potential. In a companion paper (Lee and Seo in Probab Theory Relat Fields 182:849–903, 2022), we proved the Eyring–Kramers formula for the corresponding class of metastable diffusion processes. In this article, we further develop this result by proving that a suitably timerescaled metastable diffusion process converges to a Markov chain on the deepest metastable valleys. This article is also an extension of (Rezakhanlou and Seo in https://arxiv.org/abs/1812.02069, 2018), which considered the same problem for metastable reversible diffusion processes. Our proof is based on the recently developed resolvent approach to metastability.
PubDate: 20220914

 Replica Symmetry Breaking in Dense Hebbian Neural Networks

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Abstract: Abstract Understanding the glassy nature of neural networks is pivotal both for theoretical and computational advances in Machine Learning and Theoretical Artificial Intelligence. Keeping the focus on dense associative Hebbian neural networks (i.e. Hopfield networks with polynomial interactions of even degree \(P >2\) ), the purpose of this paper is twofold: at first we develop rigorous mathematical approaches to address properly a statistical mechanical picture of the phenomenon of replica symmetry breaking (RSB) in these networks, then—deepening results stemmed via these routes—we aim to inspect the glassiness that they hide. In particular, regarding the methodology, we provide two techniques: the former (closer to mathematical physics in spirit) is an adaptation of the transport PDE to this case, while the latter (more probabilistic in its nature) is an extension of Guerra’s interpolation breakthrough. Beyond coherence among the results, either in replica symmetric and in the onestep replica symmetry breaking level of description, we prove the Gardner’s picture (heuristically achieved through the replica trick) and we identify the maximal storage capacity by a groundstate analysis in the BaldiVenkatesh highstorage regime. In the second part of the paper we investigate the glassy structure of these networks: at difference with the replica symmetric scenario (RS), RSB actually stabilizes the spinglass phase. We report huge differences w.r.t. the standard pairwise Hopfield limit: in particular, it is known that it is possible to express the free energy of the Hopfield neural network (and, in a cascade fashion, all its properties) as a linear combination of the free energies of a hard spin glass (i.e. the Sherrington–Kirkpatrick model) and a soft spin glass (the Gaussian or ”spherical” model). While this continues to hold also in the first step of RSB for the Hopfield model, this is no longer true when interactions are more than pairwise (whatever the level of description, RS or RSB). For dense networks solely the free energy of the hard spin glass survives. As the Sherrington–Kirkpatrick spin glass is fullRSB (i.e. Parisi theory holds for that model), while the Gaussian spinglass is replica symmetric, these different representation theorems prove a huge diversity in the underlying glassiness of associative neural networks.
PubDate: 20220908

 Granular Gas of Inelastic and Rough Maxwell Particles

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Abstract: Abstract The most widely used model for granular gases is perhaps the inelastic hardsphere model (IHSM), where the grains are assumed to be perfectly smooth spheres colliding with a constant coefficient of normal restitution. A much more tractable model is the inelastic Maxwell model (IMM), in which the velocitydependent collision rate is replaced by an effective meanfield constant. This simplification has been taken advantage of by many researchers to find a number of exact results within the IMM. On the other hand, both the IHSM and IMM neglect the impact of roughness—generally present in real grains—on the dynamic properties of a granular gas. This is remedied by the inelastic rough hardsphere model (IRHSM), where, apart from the coefficient of normal restitution, a constant coefficient of tangential restitution is introduced. In parallel to the simplification carried out when going from the IHSM to the IMM, we propose in this paper an inelastic rough Maxwell model (IRMM) as a simplification of the IRHSM. The tractability of the proposed model is illustrated by the exact evaluation of the collisional moments of first and second degree, and the most relevant ones of third and fourth degree. The results are applied to the evaluation of the rotationaltotranslational temperature ratio and the velocity cumulants in the homogeneous cooling state.
PubDate: 20220907

 Revisit of Macroscopic Dynamics for Some Nonequilibrium Chemical
Reactions from a Hamiltonian Viewpoint
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Abstract: Abstract Most biochemical reactions in living cells are open systems interacting with environment through chemostats to exchange both energy and materials. At a mesoscopic scale, the number of each species in those biochemical reactions can be modeled by a random timechanged Poisson processes. To characterize macroscopic behaviors in the large number limit, the law of large numbers in the path space determines a meanfield limit nonlinear reaction rate equation describing the dynamics of the concentration of species, while the WKB expansion for the chemical master equation yields a Hamilton–Jacobi equation and the Legendre transform of the corresponding Hamiltonian gives the good rate function (action functional) in the large deviation principle. In this paper, we decompose a general macroscopic reaction rate equation into a conservative part and a dissipative part in terms of the stationary solution to the Hamilton–Jacobi equation. This stationary solution is used to determine the energy landscape and thermodynamics for general chemical reactions, which particularly maintains a positive entropy production rate at a nonequilibrium steady state. The associated energy dissipation law at both the mesoscopic and macroscopic levels is proved together with a passage from the mesoscopic to macroscopic one. A nonconvex energy landscape emerges from the convex mesoscopic relative entropy functional in the large number limit, which picks up the nonequilibrium features. The existence of this stationary solution is ensured by the optimal control representation at an undetermined time horizon for the weak KAM solution to the stationary Hamilton–Jacobi equation. Furthermore, we use a symmetric Hamiltonian to study a class of nonequilibrium enzyme reactions, which leads to nonconvex energy landscape due to flux grouping degeneracy and reduces the conservative–dissipative decomposition to an Onsagertype strong gradient flow. This symmetric Hamiltonian implies that the transition paths between multiple steady states (rare events in biochemical reactions) is a modified time reversed least action path with associated path affinities and energy barriers. We illustrate this idea through a bistable catalysis reaction and compute the energy barrier for the transition path connecting two steady states via its energy landscape.
PubDate: 20220905

 Path Integral Derivation and Numerical Computation of Large Deviation
Prefactors for Nonequilibrium Dynamics Through Matrix Riccati Equations
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Abstract: Abstract For many nonequilibrium dynamics driven by small noise, in physics, chemistry, biology, or economy, rare events do matter. Large deviation theory then explains that the leading order term of the main statistical quantities have an exponential behavior. The exponential rate is often obtained as the infimum of an action, which is minimized along an instanton. In this paper, we consider the computation of the next order subexponential prefactors, which are crucial for a large number of applications. Following a path integral approach, we derive the dynamics of the Gaussian fluctuations around the instanton and compute from it the subexponential prefactors. As might be expected, the formalism leads to the computation of functional determinants and matrix Riccati equations. By contrast with the cases of equilibrium dynamics with detailed balance or generalized detailed balance, we stress the specific non locality of the solutions of the Riccati equation: the prefactors depend on fluctuations all along the instanton and not just at its starting and ending points. We explain how to numerically compute the prefactors. The case of statistically stationary quantities requires considerations of non trivial initial conditions for the matrix Riccati equation.
PubDate: 20220905

 Asymptotics for Cliques in ScaleFree Random Graphs

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Abstract: Abstract In this paper we establish asymptotics (as the size of the graph grows to infinity) for the expected number of cliques in the Chung–Lu inhomogeneous random graph model in which vertices are assigned independent weights which have tail probabilities \(h^{1\alpha }l(h)\) , where \(\alpha >2\) and l is a slowly varying function. Each pair of vertices is connected by an edge with a probability proportional to the product of the weights of those vertices. We present a complete set of asymptotics for all clique sizes and for all noninteger \(\alpha > 2\) . We also explain why the case of an integer \(\alpha \) is different, and present partial results for the asymptotics in that case.
PubDate: 20220903

 Path Large Deviations for the Kinetic Theory of Weak Turbulence

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Abstract: Abstract We consider a generic Hamiltonian system of nonlinear interacting waves with 3wave interactions. In the kinetic regime of wave turbulence, which assumes weak nonlinearity and large system size, the relevant observable associated with the wave amplitude is the empirical spectral density that appears as the natural precursor of the spectral density, or spectrum, for finite system size. Following classical derivations of the Peierls equation for the moment generating function of the wave amplitudes in the kinetic regime, we propose a large deviation estimate for the dynamics of the empirical spectral density, where the number of admissible wavenumbers, which is proportional to the volume of the system, appears as the natural large deviation parameter. The large deviation stochastic Hamiltonian that quantifies the minus of the logprobability of a trajectory is computed within the kinetic regime which assumes the Random Phase approximation for weak nonlinearity. We compare this Hamiltonian with the one for a system of modes interacting in a meanfield way with the empirical spectrum. Its relationship with the Random Phase and Amplitude approximation is discussed. Moreover, for the specific case when no forces and dissipation are present, a few fundamental properties of the large deviation dynamics are investigated. We show that the latter conserves total energy and momentum, as expected for a 3wave interacting systems. In addition, we compute the equilibrium quasipotential and check that global detailed balance is satisfied at the large deviation level. Finally, we discuss briefly some physical applications of the theory.
PubDate: 20220903

 An Asymptotic Radius of Convergence for the Loewner Equation and
Simulation of $$SLE_{\kappa }$$ S L E κ Traces via Splitting
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Abstract: Abstract In this paper, we study the convergence of Taylor approximations for the backward SLE maps near the origin. In addition, this result highlights the limitations of using stochastic Taylor methods for approximating \(SLE_{\kappa }\) traces. Due to the analytically tractable vector fields of the Loewner equation, we will show the Ninomiya–Victoir splitting is particularly well suited for SLE simulation. We believe that this is the first high order numerical method that has been successfully applied to \(SLE_{\kappa }\) .
PubDate: 20220903
