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We consider reversible ergodic Markov chains with finite state space, and we introduce a new notion of quasi-stationary distribution that does not require the presence of any absorbing state. In our setting, the hitting time of the absorbing set is replaced by an optimal strong stationary time, representing the ``hitting time of the stationary distribution''. On the one hand, we show that our notion of quasi-stationary distribution corresponds to the natural generalization of the \emph{Yaglom limit}. On the other hand, similarly to the classical quasi-stationary distribution, we show that it can be written in terms of the eigenvectors of the underlying Markov kernel, and it is therefore amenable of a geometric interpretation. Moreover, we recover the usual exponential behavior that characterizes quasi-stationary distributions and metastable systems. We also provide some toy examples, which show that the phenomenology is richer compared to the absorbing case. Finally, we present some counterexamples, showing that the assumption on the reversibility cannot be weakened in general., Comment: 20 pages, 7 figures

This is the third in a series of three papers in which we study a lattice gas subject to Kawasaki dynamics at inverse temperature $\beta>0$ in a large finite box $\Lambda_\beta \subset \mathbb{Z}^2$ whose volume depends on $\beta$. Each pair of neighbouring particles has a negative binding energy $-U<0$, while each particle has a positive activation energy $\Delta>0$. The initial configuration is drawn from the grand-canonical ensemble restricted to the set of configurations where all the droplets are subcritical. Our goal is to describe, in the metastable regime $\Delta \in (U,2U)$ and in the limit as $\beta\to\infty$, how and when the system nucleates, i.e., creates a critical droplet somewhere in $\Lambda_\beta$ that subsequently grows by absorbing particles from the surrounding gas. In the first paper we showed that subcritical droplets behave as quasi-random walks. In the second paper we used the results in the first paper to analyse how subcritical droplets form and dissolve on multiple space-time scales when the volume is moderately large, namely, $|\Lambda_\beta| = \mathrm{e}^{\theta\beta}$ with $\Delta < \theta < 2\Delta-U$. In the present paper we consider the setting where the volume is very large, namely, $|\Lambda_\beta| = \mathrm{e}^{\Theta\beta}$ with $\Delta < \Theta < \Gamma-(2\Delta-U)$, where $\Gamma$ is the energy of the critical droplet in the local model with fixed volume, and use the results in the first two papers to identify the nucleation time and the tube of typical trajectories towards nucleation. We will see that in a very large volume critical droplets appear more or less independently in boxes of moderate volume, a phenomenon referred to as homogeneous nucleation. One of the key ingredients in the proof is an estimate showing that no information can travel between these boxes on relevant time scales., Comment: 21 pages, 5 figures. arXiv admin note: text overlap with arXiv:2304.14099

Quadratic Unconstrained Binary Optimization (QUBO or UBQP) is concerned with maximizing/minimizing the quadratic form $H(J, \eta) = W \sum_{i,j} J_{i,j} \eta_{i} \eta_{j}$ with $J$ a matrix of coefficients, $\eta \in \{0, 1\}^N$ and $W$ a normalizing constant. In the statistical mechanics literature, QUBO is a lattice gas counterpart to the (generalized) Sherrington--Kirkpatrick spin glass model. Finding the optima of $H$ is an NP-hard problem. Several problems in combinatorial optimization and data analysis can be mapped to QUBO in a straightforward manner. In the combinatorial optimization literature, random instances of QUBO are often used to test the effectiveness of heuristic algorithms. Here we consider QUBO with random independent coefficients and show that if the $J_{i,j}$'s have zero mean and finite variance then, after proper normalization, the minimum and maximum \emph{per particle} of $H$ do not depend on the details of the distribution of the couplings and are concentrated around their expected values. Further, with the help of numerical simulations, we study the minimum and maximum of the objective function and provide some insight into the structure of the minimizer and the maximizer of $H$. We argue that also this structure is rather robust. Our findings hold also in the diluted case where each of the $J_{i,j}$'s is allowed to be zero with probability going to $1$ as $N \to \infty$ in a suitable way., Comment: 20 pages, 8 figures

We discuss a model describing the spin orbit resonance cascade. We assume that the primary has a two-layer (core-shell) structure: it is composed by a thin solid crust and an inner and heavier solid core that are interacting due to the presence of a fluid interface. We assume two sources of dissipation: a viscous one, depending on the relative angular velocity between core and crust and a tidal one, smaller than the first, due to the viscoelastic structure of the core. We show how these two sources of dissipation are needful for the capture in spin-orbit resonance. The crust and the core fall in resonance with different time scales if the viscous coupling between them is big enough. Finally, the tidal dissipation of the viscoelastic core, decreasing the eccentricity, brings the system out of the resonance in a third very long time scale. This mechanism of entry and exit from resonance ends in the $1:1$ stable state., Comment: 15 pages, 6 figures

We prove the equivalence between integral and local central limit theorem for spin system interacting via an absolutely summable pair potential without any conditions on the temperature of the system.

We investigate the occurrence of elements of order $p$ in the upper central series of a finite $p$-group., Comment: 11 pages - In this version we have added some indecomposable examples

In this paper we propose a simplified model to describe the dissipative effects of tides. We assume a spherical Earth with a dissipative coupling with a mechanical dumbbell. The latter has a mass much smaller than the Earth's, and it models the presence of the tidal bulges. Using properly the scale analysis, we will show that some of the consequences of tidal dissipation are the circularization and the enlargement of orbit of the Moon and the slowing down of the Earth's rotation. We will also see that tidal dissipation plays a fundamental role for the establishment of a regime of spin-orbit resonance in the celestial systems. The mathematical tools used make our treatment appropriate for senior high school students or college students., Comment: 10 pages, 3 figures

This is the second in a series of three papers in which we study a lattice gas subject to Kawasaki conservative dynamics at inverse temperature $\beta>0$ in a large finite box $\Lambda_\beta \subset\mathbb Z^2$ whose volume depends on $\beta$. Each pair of neighbouring particles has a negative binding energy $-U<0$, while each particle has a positive activation energy $\Delta>0$. The initial configuration is drawn from the grand-canonical ensemble restricted to the set of configurations where all the droplets are subcritical. Our goal is to describe, in the metastable regime $\Delta\in(U,2U)$ and in the limit as $\beta\to\infty$, how and when the system nucleates. In the first paper we showed that subcritical droplets behave as quasi-random walks. In the present paper we use the results in the first paper to analyse how subcritical droplets form and dissolve on multiple space-time scales when the volume is moderately large, i.e., $|\Lambda_\beta|=\mathrm e^{\Theta\beta}$ with $\Delta<\Theta<2\Delta-U$. In the third paper we consider the setting where the volume is very large, namely, $|\Lambda_\beta|=\mathrm e^{\Theta\beta}$ with $\Delta<\Theta<\Gamma-(2\Delta-U)$, where $\Gamma$ is the energy of the critical droplet in the local model with fixed volume, and use the results in the first two papers to identify the nucleation time. We will see that in a very large volume critical droplets appear more or less independently in boxes of moderate volume, a phenomenon referred to as homogeneous nucleation. Since Kawasaki dynamics is conservative, i.e., particles are preserved, we need to control non-local effects in the way droplets are formed and dissolved. This is done via a deductive approach: the tube of typical trajectories leading to nucleation is described via a series of events on which the evolution of the gas consists of droplets wandering around on multiple space-time scales., Comment: 46 pages, 11 figures

We analyse the Blume-Emery-Griffiths (BEG) model on the lattice $\Zd$ at the ferromagnetic-antiquadrupolar-disordered (FAD) and antiquadrupolar-disordered (AD) interfaces of parameters. In our analysis of the FAD interface we introduce a Gibbs sampler of the ground states at zero temperature, and we exploit it in two different ways: first, we perform via perfect sampling an empirical evaluation of the spontaneous magnetization at zero temperature, finding a non-zero value in $d=3$ and a vanishing value in $d=2$. Second, using a careful coupling with the Bernoulli site percolation model in $d=2$, we prove rigorously that imposing $+$ boundary conditions, the magnetization in the center of a square box tends to zero in the thermodynamical limit and the two-point correlations decay exponentially. Also, using again a coupling argument, we show that the infinite volume Gibbs measure of the zero-temperature BEG exists and it is unique. In our analysis of the AD interface we restrict ourselves to $d=2$ and, by comparing the BEG model with a Bernoulli site percolation in a matching graph of $\mathbb{Z}^2$, we get a condition for the vanishing of the infinite volume limit magnetization improving, for low temperatures, earlier results obtained via expansion techniques.

In a recent paper on a study of the Sylow 2-subgroups of the symmetric group with 2^n elements it has been show that the growth of the first (n-2) consecutive indices of a certain normalizer chain is linked to the sequence of partitions of integers into distinct parts. Unrefinable partitions into distinct parts are those in which no part x can be replaced with integers whose sum is x obtaining a new partition into distinct parts. We prove here that the (n-1)-th index of the previously mentioned chain is related to the number of unrefinable partitions into distinct parts satisfying a condition on the minimal excludant.

In recent years, we have come to appreciate the astounding intricacy of the formation process of minerals from ions in aqueous solutions. In this context, a number of studies have revealed that nucleation in the calcium sulfate system is non-classical, involving the aggregation and reorganization of nanosized prenucleation particles. In a recent work we have shown that this particle-mediated nucleation pathway is actually imprinted in the resultant single micron-sized CaSO4 crystals. This property of CaSO4 minerals provides us with an unique opportunity to search for evidence of non-classical nucleation pathways in geological environments. In particular, we focused on the quintessential single crystals of anhydrite extracted from the Naica mine in Mexico. We elucidated the growth history from this mineral sample by mapping growth defects at different length scales. Based on these data we argue that the nano-scale misalignment of the structural sub-units observed in the initial calcium sulfate crystal seed propagate through different length-scales both in morphological, as well as strictly crystallographic aspects, eventually causing the formation of large mesostructured single crystals of anhydrite. Hence, the nanoparticle mediated nucleation mechanism introduces a 'seed of imperfection', which leads to a macroscopic single crystal, in which its fragments do not fit together at different length-scales in a self-similar manner. Consequently, anisotropic voids of various sizes are formed with very well-defined walls/edges. But, at the same time the material retains its essential single crystal nature. These findings shed new light on the longstanding concept of crystal structure., Comment: The dataset and SI files are published here: https://doi.org/10.5281/zenodo.4943234

On the space of $\pm 1$ spin configurations on the 3$d$-square lattice, we consider the \emph{shaken dynamics}, a parallel Markovian dynamics that can be interpreted in terms of Probabilistic Cellular Automata. The transition probabilities are defined in terms of pair ferromagnetic Ising-type Hamiltonians with nearest neighbor interaction $J$, depending on an additional parameter $q$, measuring the tendency of the system to remain locally in the same state. Odd times and even times have different transition probabilities. We compute the stationary measure of the shaken dynamics and we investigate its relation with the Gibbs measure for the 3$d$ Ising model. It turns out that the two parameters $J$ and $q$ tune the geometry of the underlying lattice. We conjecture the existence of unique line of critical points in $J-q$ plane. By a judicious use of perturbative methods we delimit the region where such curve must lie and we perform numerical simulation to determine it. Our method allows us to find in a unified way the critical values of $J$ for Ising model with first neighbors interaction, defined on a whole class of lattices, intermediate between the two-dimensional hexagonal and the three-dimensional cubic one, such as, for example, the tetrahedral lattice. Finally we estimate the critical exponents of the magnetic susceptibility and show that our model captures a dimensional transition in the geometry of the system at $q = 0$.

In this short note we present an alternative proof of the so-called First Visit Time Lemma (FVTL), originally presented by Cooper and Frieze in its first formulation in [21], and then used and refined in a list of papers by Cooper, Frieze and coauthors. We work in the original setting, considering a growing sequence of irreducible Markov chains on $n$ states. We assume that the chain is rapidly mixing and with a stationary measure having no entry which is too small nor too large. Under these assumptions, the FVTL shows the exponential decay of the distribution of the hitting time of a given state $x$ -- for the chain started at stationarity -- up to a small multiplicative correction. While the proof of the FVTL presented by Cooper and Frieze is based on tools from complex analysis, and it requires an additional assumption on a generating function, we present a completely probabilistic proof, relying on the theory of quasi-stationary distributions and on strong-stationary times arguments. In addition, under the same set of assumptions, we provide some quantitative control on the Doob's transform of the chain on the complement of the state $x$.

We discuss a model describing the effects of tidal dissipation on satellite's orbits. Tidal bulges are described in terms of a dumbbell, coupled to the rotation by a dissipative interaction. The assumptions on this dissipative coupling turns out to be crucial in the evolution of the system., Comment: 2 figures

On the space of Ising configurations on the 2-d square lattice, we consider a family of non Gibbsian measures introduced by using a pair Hamiltonian, depending on an additional inertial parameter $q$. These measures are related to the usual Gibbs measure on $\Z^2$ and turn out to be the marginal of the Gibbs measure of a suitable Ising model on the hexagonal lattice. The inertial parameter $q$ tunes the geometry of the system. The critical behaviour and the decay of correlation functions of these measures are studied thanks to relation with the Random Cluster model.

We define a class of Markovian parallel dynamics for spin systems on arbitrary graphs with nearest neighbor interaction described by a Hamiltonian function $H(\sigma)$. These dynamics turn out to be reversible and their stationary measure is explicitly determined. Convergence to equilibrium and relation of the stationary measure to the usual Gibbs measure are discussed when the dynamics is defined on $\mathbb{Z}^2$. Further it is shown how these dynamics can be used to define natively parallel algorithms to face problems in the context of combinatorial optimization., Comment: 8 figures

The novel notion of rigid commutators is introduced to determine the sequence of the logarithms of the indices of a certain normalizer chain in the Sylow 2-subgroup of the symmetric group on 2^n letters. The terms of this sequence are proved to be those of the partial sums of the partitions of an integer into at least two distinct parts, that relates to a famous Euler's partition theorem.

On the basis of an initial interest in symmetric cryptography, in the present work we study a chain of subgroups. Starting from a Sylow $2$-subgroup of AGL(2,n), each term of the chain is defined as the normalizer of the previous one in the symmetric group on $2^n$ letters. Partial results and computational experiments lead us to conjecture that, for large values of $n$, the index of a normalizer in the consecutive one does not depend on $n$. Indeed, there is a strong evidence that the sequence of the logarithms of such indices is the one of the partial sums of the numbers of partitions into at least two distinct parts.

In this paper we propose a notion of stability, that we call $\epsilon -N$-stability, for systems of particles interacting via Newton's gravitational potential, and orbiting a much bigger object. For these systems the usual thermodynamical stability condition, ensuring the possibility to perform the thermodynamical limit, fails, but one can use as relevant parameter the maximum number of particles $N$ that guarantees the $\epsilon -N$-stability. With some judicious but not particularly optimized estimates, borrowed from the classical theory of equilibrium statistical mechanics, we show that our model has a good fit with the data observed in the Solar System, and it gives a reasonable interpretation of some of its global properties.

We present a simple one-dimensional Ising-type spin system on which we define a completely asymmetric Markovian single spin-flip dynamics. We study the system at a very low, yet non-zero, temperature and we show that for empty boundary conditions the Gibbs measure is stationary for such dynamics, while introducing in a single site a $+$ condition the stationary measure changes drastically, with macroscopical effects. We achieve this result defining an absolutely convergent series expansion of the stationary measure around the zero temperature system. Interesting combinatorial identities are involved in the proofs.

In this paper we study the relationships between the elementary abelian regular subgroups and the Sylow $2$-subgroups of their normalisers in the symmetric group $\mathrm{Sym}(\mathbb{F}_2^n)$, in view of the interest that they have recently raised for their applications in symmetric cryptography.

In the setting of non-reversible Markov chains on finite or countable state space, exact results on the distribution of the first hitting time to a given set $G$ are obtained. A new notion of "strong metastability time" is introduced to describe the local relaxation time. This time is defined via a generalization of the strong stationary time to a "conditionally strong quasi-stationary time"(CSQST). Rarity of the target set $G$ is not required and the initial distribution can be completely general. The results clarify the the role played by the initial distribution on the exponential law; they are used to give a general notion of metastability and to discuss the relation between the exponential distribution of the first hitting time and metastability.

We construct a parallel stochastic dynamics with invariant measure converging to the Gibbs measure of the low temperature Ising model. The proof of such convergence requires a polymer expansion based on suitably defined Peierls-type contours.

We study rigorously a lattice gas version of the Sherrington-Kirckpatrick spin glass model. In discrete optimization literature this problem is known as Unconstrained Binary Quadratic Programming (UBQP) and it belongs to the class NP-hard. We prove that the fluctuations of the ground state energy tend to vanish in the thermodynamic limit, and we give a lower bound of such ground state energy. Then we present an heuristic algorithm, based on a probabilistic cellular automaton, which seems to be able to find configurations with energy very close to the minimum, even for quite large instances., Comment: 3 figures, 2 tables