arXiv subjects

Miquel Montero

Publications and source records attributed to Miquel Montero.

At least 19 recordsLinked to original sources

arXiv2025

Emergence of social hierarchies in a society with two competitive groups

Agent-based models describing social interactions among individuals can help to better understand emerging macroscopic patterns in societies. One of the topics which is worth tackling is the formation of different kinds of hierarchies that emerge in social spaces such as cities. Here we propose a Bonabeau-like model by adding a second group of agents. The fundamental particularity of our model is that only a pairwise interaction between agents of the opposite group is allowed. Agent fitness can thus only change by competition among the two groups, while the total fitness in the society remains constant. The main result is that for a broad range of values of the model parameters, the fitness of the agents of each group show a decay in time except for one or very few agents which capture almost all the fitness in the society. Numerical simulations also reveal a singular shift from egalitarian to hierarchical society for each group. This behaviour depends on the control parameter $η$, playing the role of the inverse of the temperature of the system. Results are invariant with regard to the system size, contingent solely on the quantity of agents within each group. Finally, scaling laws are provided thus showing a data collapse from different model parameters and they follow a shape which can be related to the presence of a phase transition in the model.

physics.soc-ph↗

arXiv2023

Discounting the distant future: What do historical bond prices imply about the long term discount rate?

We present a thorough empirical study on real interest rates by also including risk aversion through the introduction of the market price of risk. With the view of complex systems science and its multidisciplinary approach, we use the theory of bond pricing to study the long term discount rate. Century-long historical records of 3 month bonds, 10 year bonds, and inflation allow us to estimate real interest rates for the UK and the US. Real interest rates are negative about a third of the time and the real yield curves are inverted more than a third of the time, sometimes by substantial amounts. This rules out most of the standard bond pricing models, which are designed for nominal rates that are assumed to be positive. We therefore use the Ornstein-Uhlenbeck model which allows negative rates and gives a good match to inversions of the yield curve. We derive the discount function using the method of Fourier transforms and fit it to the historical data. The estimated long term discount rate is $1.7$ \% for the UK and $2.2$ \% for the US. The value of $1.4$ \% used by Stern is less than a standard deviation from our estimated long run return rate for the UK, and less than two standard deviations of the estimated value for the US. All of this once more reinforces the support for immediate and substantial spending to combat climate change.

q-fin.MF↗

arXiv2023

The effect of stochastic resettings on the counting of level crossings for inertial random processes

We study the counting of level crossings for inertial random processes exposed to stochastic resetting events. We develop the general approach of stochastic resetting for inertial processes with sudden changes in the state characterized by position and velocity. We obtain the level-crossing intensity in terms of that of underlying reset-free process, for resetting events with Poissonian statistics. We apply this result to the random acceleration process and the inertial Brownian motion. In both cases, we show that there is an optimal resetting rate that maximizes the crossing intensity, and we obtain the asymptotic behavior of the crossing intensity for large and small resetting rates. Finally, we discuss the stationary distribution and the mean first-arrival time in the presence of resetting.

cond-mat.stat-mech↗

arXiv2023

Merge of two oppositely biased Wiener processes

We introduce a technique to merge two biased Brownian motions into a single regular process. The outcome follows a stochastic differential equation with a constant diffusion coefficient and a non-linear drift. The emerging stochastic process has outstanding properties, such as spatial and temporal translational invariance of its mean squared displacement, and can be efficiently simulated via a random walk with site-dependent one-step transition probabilities.

math.PR↗

arXiv2020

Statistical analysis and stochastic interest rate modelling for valuing the future with implications in climate change mitigation

High future discounting rates favor inaction on present expending while lower rates advise for a more immediate political action. A possible approach to this key issue in global economy is to take historical time series for nominal interest rates and inflation, and to construct then real interest rates and finally obtaining the resulting discount rate according to a specific stochastic model. Extended periods of negative real interest rates, in which inflation dominates over nominal rates, are commonly observed, occurring in many epochs and in all countries. This feature leads us to choose a well-known model in statistical physics, the Ornstein-Uhlenbeck model, as a basic dynamical tool in which real interest rates randomly fluctuate and can become negative, even if they tend to revert to a positive mean value. By covering 14 countries over hundreds of years we suggest different scenarios and include an error analysis in order to consider the impact of statistical uncertainty in our results. We find that only 4 of the countries have positive long-run discount rates while the other ten countries have negative rates. Even if one rejects the countries where hyperinflation has occurred, our results support the need to consider low discounting rates. The results provided by these fourteen countries significantly increase the priority of confronting global actions such as climate change mitigation. We finally extend the analysis by first allowing for fluctuations of the mean level in the Ornstein-Uhlenbeck model and secondly by considering modified versions of the Feller and lognormal models. In both cases, results remain basically unchanged thus demonstrating the robustness of the results presented.

q-fin.MF↗

arXiv2019

Random walk with hyperbolic probabilities

The random walk with hyperbolic probabilities that we are introducing is an example of stochastic diffusion in a one-dimensional heterogeneous media. Although driven by site-dependent one-step transition probabilities, the process retains some of the features of a simple random walk, shows other traits that one would associate with a biased random walk and, at the same time, presents new properties not related with either of them. In particular, we show how the system is not fully ergodic, as not every statistic can be estimated from a single realization of the process. We further give a geometric interpretation for the origin of these irregular transition probabilities.

cond-mat.stat-mech↗

arXiv2019

Escape probabilities of compound renewal processes with drift

We consider the problem of determining escape probabilities from an interval of a general compound renewal process with drift. This problem is reduced to the solution of a certain integral equation. In an actuarial situation where only negative jumps arise we give a general solution for escape and survival probabilities under Erlang$(n)$ and hypo-exponential arrivals. These ideas are generalized to the class of arrival distributions having rational Laplace transforms. In a general situation with two-sided jumps we also identify important families of solvable cases. A parallelism with the "scale function" of diffusion processes is drawn.

math.PR↗

arXiv2017

Continuous-time random walks with reset events: Historical background and new perspectives

In this paper, we consider a stochastic process that may experience random reset events which relocate the system to its starting position. We focus our attention on a one-dimensional, monotonic continuous-time random walk with a constant drift: the process moves in a fixed direction between the reset events, either by the effect of the random jumps, or by the action of a deterministic bias. However, the orientation of its motion is randomly determined after each restart. As a result of these alternating dynamics, interesting properties do emerge. General formulas for the propagator as well as for two extreme statistics, the survival probability and the mean first-passage time, are also derived. The rigor of these analytical results is verified by numerical estimations, for particular but illuminating examples.

math-ph↗

arXiv2017

Quantum and random walks as universal generators of probability distributions

Quantum walks and random walks bear similarities and divergences. One of the most remarkable disparities affects the probability of finding the particle at a given location: typically, almost a flat function in the first case and a bell-shaped one in the second case. Here I show how one can impose any desired stochastic behavior (compatible with the continuity equation for the probability function) on both systems by the appropriate choice of time- and site-dependent coins. This implies, in particular, that one can devise quantum walks that show diffusive spreading without loosing coherence, as well as random walks that exhibit the characteristic fast propagation of a quantum particle driven by a Hadamard coin.

quant-ph↗

arXiv2016

Sisyphus random walk

In this paper we consider a particular version of the random walk with restarts: random reset events which bring suddenly the system to the starting value. We analyze its relevant statistical properties like the transition probability and show how an equilibrium state appears. Formulas for the first-passage time, high-water marks and other extreme statistics are also derived: we consider counting problems associated naturally to the system. Finally we indicate feasible generalizations useful for interpreting different physical effects.

math-ph↗

arXiv2016

Invariance in Quantum Walks

In this Chapter, we present some interesting properties of quantum walks on the line. We concentrate our attention in the emergence of invariance and provide some insights into the ultimate origin of the observed behavior. In the first part of the Chapter, we review the building blocks of the quantum-mechanical version of the standard random walk in one dimension. The most distinctive difference between random and quantum walks is the replacement of the random coin in the former by the action of a unitary operator upon some internal property of the later. We provide explicit expressions for the solution to the problem when the most general form for the homogeneous unitary operator is considered, and we analyze several key features of the system as the presence of symmetries or stationary limits. After that, we analyze the consequences of letting the properties of the coin operator change from site to site, and from time step to time step. In spite of this lack of homogeneity, the probabilistic properties of the motion of the walker can remain unaltered if the coin variability is chosen adequately. Finally, we show how this invariance can be connected to the gauge freedom of electromagnetism.

quant-ph↗

arXiv2016

Classical-like behavior in quantum walks with inhomogeneous, time-dependent coin operators

Although quantum walks exhibit peculiar properties that distinguish them from random walks, classical behavior can be recovered in the asymptotic limit by destroying the coherence of the pure state associated to the quantum system. Here I show that this is not the only way: I introduce a quantum walk driven by an inhomogeneous, time-dependent coin operator, which mimics the statistical properties of a random walk at all time scales. The quantum particle undergoes unitary evolution and, in fact, the high correlation evidenced by the components of the wave function can be used to revert the outcome of an accidental measurement of its chirality.

quant-ph↗

arXiv2014

Invariance in quantum walks with time-dependent coin operators

In this paper we unveil some features of a discrete-time quantum walk on the line whose coin depends on the temporal variable. After considering the most general form of the unitary coin operator, we focus on the role played by the two phase factors that one can incorporate there, and show how both terms influence the evolution of the system. A closer analysis reveals that the probabilistic properties of the motion of the walker remain unaltered when the update rule of these phases is chosen adequately. This invariance is based on a symmetry with consequences not yet fully explored.

quant-ph↗

arXiv2014

Quantum walk with a general coin: Exact solution and asymptotic properties

In this paper we present closed-form expressions for the wave function that governs the evolution of the discrete-time quantum walk on a line when the coin operator is arbitrary. The formulas were derived assuming that the walker can either remain put in the place or proceed in a fixed direction but never move backward, although they can be easily modified to describe the case in which the particle can travel in both directions. We use these expressions to explore the properties of magnitudes associated to the process, as the probability mass function or the probability current, even though we also consider the asymptotic behavior of the exact solution. Within this approximation, we will estimate upper and lower bounds, consider the origins of an emerging approximate symmetry, and deduce the general form of the stationary probability density of the relative location of the walker.

quant-ph↗

arXiv2013

Uncertain growth and the value of the future

For environmental problems such as global warming future costs must be balanced against present costs. This is traditionally done using an exponential function with a constant discount rate, which reduces the present value of future costs. The result is highly sensitive to the choice of discount rate and has generated a major controversy as to the urgency for immediate action. We study analytically several standard interest rate models from finance and compare their properties to empirical data. From historical time series for nominal interest rates and inflation covering 14 countries over hundreds of years, we find that extended periods of negative real interest rates are common, occurring in many epochs in all countries. This leads us to choose the Ornstein-Uhlenbeck model, in which real short run interest rates fluctuate stochastically and can become negative, even if they revert to a positive mean value. We solve the model in closed form and prove that the long-run discount rate is always less than the mean; indeed it can be zero or even negative, despite the fact that the mean short term interest rate is positive. We fit the parameters of the model to the data, and find that nine of the countries have positive long run discount rates while five have negative long-run discount rates. Even if one rejects the countries where hyperinflation has occurred, our results support the low discounting rate used in the Stern report over higher rates advocated by others.

q-fin.ST↗

arXiv2013

Unidirectional quantum walks: evolution and exit times

In this paper we focus our attention on a particle that follows a unidirectional quantum walk, an alternative version of the nowadays widespread discrete-time quantum walk on a line. Here the walker at each time step can either remain in place or move in a fixed direction, e.g., rightward or upward. While both formulations are essentially equivalent, the present approach leads to consider Discrete Fourier Transforms, which eventually results in obtaining explicit expressions for the wave functions in terms of finite sums, and allows the use of efficient algorithms based on the Fast Fourier Transform. The wave functions here obtained govern the probability of finding the particle at any given location, but determine as well the exit-time probability of the walker from a fixed interval, which is also analyzed.

quant-ph↗

arXiv2012

Monotonous continuous-time random walks with drift and stochastic reset events

In this paper we consider a stochastic process that may experience random reset events which bring suddenly the system to the starting value and analyze the relevant statistical magnitudes. We focus our attention on monotonous continuous-time random walks with a constant drift: the process increases between the reset events, either by the effect of the random jumps, or by the action of the deterministic drift. As a result of all these combined factors interesting properties emerge, like the existence|for any drift strength|of a stationary transition probability density function, or the faculty of the model to reproduce power-law-like behavior. General formulas for two extreme statistics, the survival probability and the mean exit time, are also derived. To corroborate in an independent way the results of the paper, Monte Carlo methods were used. These numerical estimations are in full agreement with the analytical predictions.

math-ph↗

arXiv2011

Parrondo-like behavior in continuous-time random walks with memory

The Continuous-Time Random Walk (CTRW) formalism can be adapted to encompass stochastic processes with memory. In this article we will show how the random combination of two different unbiased CTRWs can give raise to a process with clear drift, if one of them is a CTRW with memory. If one identifies the other one as noise, the effect can be thought as a kind of stochastic resonance. The ultimate origin of this phenomenon is the same of the Parrondo's paradox in game theory

math-ph↗