Roberto Mulet

The longest increasing subsequence (LIS) problem—a classic combinatorial challenge with deep connections to statistical mechanics—exhibits a rich thermodynamic landscape. Introducing a temperature, we identify two distinct energy scales: a Schottky-like crossover at T cross and a condensation transition at T cond , below which the number of maximum-length configurations becomes sub-exponential in system size. We also show that despite the existence of polynomial-time dynamic programming algorithms for the ground state, local Monte Carlo dynamics, after sudden quenches at low temperatures, become trapped in metastable state configuration displaying characteristic glassy signatures: two-step relaxation, persistent dynamical overlaps, and aging. On the other hand, logarithmic annealing tracks equilibrium down to the ground state. These results establish that thermodynamic sparsity—not energetic barriers—can render local search dynamically intractable, positioning the LIS problem as a bridge between exactly solvable optimization and glassy spin-glass phenomenology.

In the past decades, many efforts have focused on analyzing typical-case hardness in optimization and inference problems. Some recent work has pointed out that polynomial algorithms exist, running with a time that grows more than linearly with the system size, which can do better than linear algorithms, finding solutions to random problems in a wider range of parameters. However, a theory for polynomial and superlinear algorithms is in general lacking. Here, we examine the performance of the simulated annealing algorithm, a standard, versatile, and robust choice for solving optimization and inference problems, in the prototypical random K -SAT problem. For the first time, we show that the algorithmic thresholds depend on the time scaling of the algorithm with the size of the system. Indeed, one can identify not just one but different thresholds for linear, quadratic, and cubic regimes (and so on). This observation opens new directions in studying the typical case hardness in optimization problems.

Recent advances in artificial intelligence have relied heavily on increasingly large neural networks, raising concerns about their computational and environmental costs. This paper investigates whether simpler, sparser networks can maintain strong performance by studying restricted Boltzmann machines (RBMs) under extreme pruning conditions. Inspired by the lottery ticket hypothesis, we demonstrate that RBMs can achieve high-quality generative performance even when up to 80% of the connections are pruned before training, confirming that they contain viable sub-networks. However, our experiments reveal crucial limitations: trained networks cannot fully recover lost performance through retraining once additional pruning is applied. We identify a sharp transition above which the generative quality degrades abruptly when pruning disrupts a minimal core of essential connections. Moreover, re-trained networks remain constrained by the parameters originally learned performing worse than networks trained from scratch at equivalent sparsity levels. These results suggest that for sparse networks to work effectively, pruning should be implemented early in training rather than attempted afterwards. Our findings provide practical insights for the development of efficient neural architectures and highlight the Persistent influence of initial conditions on network capabilities.

We solve exactly the non-equilibrium dynamics of two discrete random walkers moving in channels with transition rates p ≠ q that swap positions at a rate s . We compute exactly the joint probability distribution P n , m ( t ) for the walkers, revealing the existence of two dynamical crossovers. The first signals the passage from independent diffusion to a swap-dominated regime, where the particles act as identical random walkers swapping positions. The second crossover occurs when both channels become indistinguishable and the walkers move around the same position. Furthermore, we demonstrate the existence of a persistent spatial anisotropy defined by the difference between the variances of the probability distributions in the two channels. Our results may provide a quantitative framework to understand diverse systems. In biology, it is motivated by motor proteins (kinesin/dynein) exchanging cargo leadership, membrane receptors swapping binding partners, or brain synapses with activity-dependent plasticity. In finance, it models traders with distinct risk profiles swapping positions in limit order books, or volatility spillover between coupled markets. These diverse systems share a unifying theme: exchange processes mediate macroscopic correlations despite individual heterogeneity.

Cell cultures exhibit rich and complex behaviors driven by dynamic metabolic interactions among cells. In this work, we present a model that captures these interactions through a framework inspired by statistical mechanics. Using Monte Carlo simulations, we explore the equilibrium and dynamical properties of a population of cells arranged in a two-dimensional lattice, where each cell is characterized by fluxes of three reactions: glucose consumption ( g ) , respiration ( r ) , and waste production/absorption ( w ) . The system minimizes an energy function influenced by competitive ( J g > 0 ) and cooperative ( J w < 0 ) couplings between cells. Our results reveal three distinct phases: a competitive phase dominated by glucose competition, a cooperative phase marked by ordered waste exchange, and a disordered phase with local-scale cooperation. By incorporating evolutionary dynamics, we demonstrate how initially noninteracting cells can develop effective metabolic interactions, leading to heterogeneous cultures sustained by cross feeding. These findings are further supported by analytical solutions derived using mean-field approximations. The model provides insights into how environmental constraints and stochastic fluctuations shape community structures, offering a versatile approach to study several emergent phenomena in biological systems.

This study pioneers the application of the market microstructure framework to an informal financial market. By scraping data from websites and social media about the Cuban informal currency market, we model the dynamics of bid/ask intentions using a Limit Order Book (LOB). This approach enables us to study key characteristics such as liquidity, stability and volume profiles. We continue exploiting the Avellaneda-Stoikov model to explore the impact of introducing a Market Maker (MM) into this informal setting, assessing its influence on the market structure and the bid/ask dynamics. We show that the Market Maker improves the quality of the market. Beyond their academic significance, we believe that our findings are relevant for policymakers seeking to intervene informal markets with limited resources.

Human Embryonic Kidney cells (HEK293) are a popular host for recombinant protein expression and production in the biotechnological industry. This has driven within both, the scientific and the engineering communities, the search for strategies to increase their protein productivity. The present work is inserted into this search exploring the impact of adding sodium acetate (NaAc) into a batch culture of HEK293 cells. We monitored, as a function of time, the cell density, many external metabolites, and the supernatant concentration of the heterologous extra-cellular domain ECD-Her1 protein, a protein used to produce a candidate prostate cancer vaccine. We observed that by adding different concentrations of NaAc (0, 4, 6 and 8 mM), the production of ECD-Her1 protein increases consistently with increasing concentration, whereas the carrying capacity of the medium decreases. To understand these results we exploited a combination of experimental and computational techniques. Metabolic Flux Analysis (MFA) was used to infer intracellular metabolic fluxes from the concentration of external metabolites. Moreover, we measured independently the extracellular acidification rate and oxygen consumption rate of the cells. Both approaches support the idea that the addition of NaAc to the culture has a significant impact on the metabolism of the HEK293 cells and that, if properly tuned, enhances the productivity of the heterologous ECD-Her1 protein.