@article{22307, abstract = {We investigate magnetic active matter in confined geometries using both experiments with magnetic toy robots, Hexbugs, and simulations of elongated magnetic active Brownian particles in circular domains. Standard active particles tend to accumulate at boundaries, forming clusters even at relatively low densities. In the presence of magnetic interactions, we provide evidence for a effect that inhibits clustering and shifts its onset to higher packing fractions. Moreover, magnetic dipolar interactions give rise to collective behaviors such as train-like formations, rotating pairs, and rotating clusters.}, author = {Musacchio, Marco and Felber, Markus and Paoluzzi, Matteo and Gnoli, Andrea and Puglisi, Andrea and Angelani, Luca}, issn = {2470-0053}, journal = {Physical Review E}, number = {5}, publisher = {American Physical Society}, title = {{Fluidization induced by magnetic interactions in confined active matter}}, doi = {10.1103/hylm-ljlf}, volume = {113}, year = {2026}, } @inproceedings{22302, abstract = {Speculative generation has emerged as a promising technique to accelerate inference in large language models (LLMs) by leveraging parallelism to verify multiple draft tokens simultaneously. However, the fundamental limits on the achievable speedup remain poorly understood. In this work, we establish the first “tight” lower bounds on the runtime of any deterministic speculative generation algorithm. This is achieved by drawing a parallel between the token generation process and branching random walks, which allows us to analyze the optimal draft tree selection problem. We prove, under basic assumptions, that the expected number of tokens successfully predicted per speculative iteration is bounded as \mathbb{E}[X] ≤ (𝜇 + 𝜇(2))log(B )/𝜇2 + O(1), where B is the verifier’s batch size, 𝜇 is the expected entropy of the verifier’s output distribution, and 𝜇(2) is this entropy’s second moment. This result provides new insights into the limits of parallel token generation, and could guide the design of future speculative decoding systems. Empirical evaluations on Llama models validate our theoretical predictions, confirming the tightness of our bounds in practical settings.}, author = {Pankratov, Sergei and Alistarh, Dan-Adrian}, booktitle = {Proceedings of the 19th Conference of the European Chapter of the Association for Computational Linguistics}, location = {Rabat, Morocco}, pages = {6404–6418}, publisher = {Association for Computational Linguistics}, title = {{Speculative decoding speed-of-light: Optimal lower bounds via branching random walks}}, doi = {10.18653/v1/2026.eacl-long.301}, year = {2026}, } @unpublished{22276, abstract = {Tissue tension is a key determinant of tissue shape, and its regulation is essential for both morphogenesis and the maintenance of tissue integrity. During zebrafish embryogenesis, the enveloping layer (EVL) – an epithelial monolayer covering the blastoderm – undergoes extensive spreading that is driven by pulling forces exerted at its margin and more than doubles its surface area. Yet whether and how the EVL actively regulates its tissue tension during this process remains unclear. Here, we show that the EVL maintains constant tissue tension while spreading, and that it achieves this by reducing apical cell contractility in response to the same pulling forces that drive its spreading. We identify a mechanosensitive pathway underlying this response, mediated by the scaffold/adaptor protein Kibra regulating the activity of atypical protein kinase C (aPKC) at the apical domain of EVL cells. Under low mechanical stretch, Kibra forms condensates at the base of actin-based apical projections, where it activates Myosin II to increase apical contractility through aPKC downregulation. As mechanical stretch increases, apical projections disassemble, Kibra condensates dissolve, and aPKC activity rises. Elevated aPKC activity in turn reduces apical contractility by reducing Myosin II activity, thereby maintaining constant tissue tension despite increased mechanical stretch. Together, these findings reveal a mechanosensitive mechanism that enables robust adaptation of tissue tension to changing mechanical stretch, ensuring efficient tissue spreading and morphogenesis.}, author = {Hino, Naoya and Kapoor, Tushna and Gubbala, Uday R and Hannezo, Edouard B and Heisenberg, Carl-Philipp J}, keywords = {Epithelial spreading, tissue tension, mechanosensation, aPKC, Kibra, zebrafish}, publisher = {Institute of Science and Technology Austria}, title = {{Apical domain mechanosensation regulates tissue tension homeostasis}}, year = {2026}, } @misc{21864, author = {Anonymous, 1 and Anonymous, 2 and Anonymous, 3}, issn = {2664-1690}, pages = {32}, publisher = {Institute of Science and Technology Austria}, title = {{Mechanism of tissue tension homeostasis during embryogenesis}}, year = {2026}, } @article{22308, abstract = {This article is a collection of contributions from speakers at the 2025 DEAMN workshop at the Majorana Centre in Erice. Not ordinary contributions to a conference proceeding, this gives a new and different perspective on the work done by the workshop participants.}, author = {Hansen, Klavs and Kresin, Vitaly and Al Hyder, Ragheed and Lemeshko, Mikhail and Fárník, Michal and Fedor, Juraj and Ferrari, Piero and Worutowicz, Laura X. and Louwerse, Rick J. and Kiawi, Denis and Waters, Laurens B. F. M. and Lang, Sandra M. and Bakker, Joost M. and von Issendorff, Bernd and Kong, Wei and Mehmel, Jannik and Schäfer, Rolf and Pedalino, Sebastian and Ramírez-Galindo, Bruno E. and Ferstl, Richard and Sindelar, Severin and Gerlich, Stefan and Arndt, Markus and Sayres, Scott G. and Wang, Lai-Sheng}, issn = {1434-6079}, journal = {The European Physical Journal D}, number = {5}, publisher = {Springer Nature}, title = {{Reflections on future problems in cluster science}}, doi = {10.1140/epjd/s10053-026-01126-x}, volume = {80}, year = {2026}, } @article{22167, abstract = {Given a vertex-ordered graph G, the ordered Ramsey number r<(G) is the minimum integer N such that every 2-coloring of the edges of the complete ordered graph KN contains a monochromatic ordered copy of G. Motivated by a similar question posed by Erd˝os and Graham [On partition theorems for finite graphs, Infinite and finite sets (Colloq., Keszthely, 1973), North-Holland, Amsterdam-London, pp. 515–527] in the unordered setting, we study the problem of bounding the ordered Ramsey number of any ordered graph G with m edges and no isolated vertices. We prove that r<(G) ≤ e109√m(log log m)3/2 for any such G, which is tight up to the (log log m)3/2 factor in the exponent. As a corollary, we obtain the corresponding bound for the oriented Ramsey number of a directed graph with m edges.}, author = {Bradač, Domagoj and Morawski, Patryk and Sudakov, Benny and Wigderson, Yuval}, issn = {1088-6826}, journal = {Proceedings of the American Mathematical Society}, number = {3}, pages = {927--942}, publisher = {American Mathematical Society}, title = {{Ordered Ramsey numbers of graphs with 𝑚 edges}}, doi = {10.1090/proc/17442}, volume = {154}, year = {2026}, } @article{22183, abstract = {A partition of a (hyper)graph is ε-homogeneous if the edge densities between almost all clusters are either at most ε or at least 1 − ε. Suppose a 3-graph has the property that the link of every vertex has an ε-homogeneous partition of size poly(1/ε). Does this guarantee that the 3-graph also has a small homogeneous partition? Terry and Wolf proved that such a 3-graph has an ε-homogeneous partition of size given by a wowzer-type function. Terry recently improved this to a double exponential bound, and conjectured that this bound is tight. Our first result in this paper disproves this conjecture by giving an improved (single) exponential bound, which is best possible. We further obtain an analogous result for k-graphs of all uniformities k 3. The above problem is part of a much broader programme, which seeks to understand the conditions under which a (hyper)graph has small ε-regular partitions. While this problem is fairly well understood for graphs, the situation is (as always) much more involved already for 3-graphs. For example, it is natural to ask if one can strengthen our first result by only requiring each link to have ε-regular partitions of size poly(1/ε). Our second result shows that surprisingly the answer is “no”, namely, a 3-graph might only have regular partitions of tower-type size, even though the link of every vertex has an ε-regular partition of polynomial size.}, author = {Gishboliner, Lior and Shapira, Asaf and Wigderson, Yuval}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, number = {4}, publisher = {Oxford University Press}, title = {{Is it easy to regularize a hypergraph with easy links?}}, doi = {10.1093/imrn/rnag018}, volume = {2026}, year = {2026}, } @article{22184, abstract = {Ramsey's theorem states that if N is sufficiently large, then no matter how one colors the edges among N vertices with two colors, there are always k vertices spanning edges in only one color. Given this theorem, it is natural to ask "how large is sufficiently large?" Ramsey's original proof showed that N=k! is sufficient, and five years later Erdős and Szekeres improved this bound to N=4^k . And then progress stalled for almost 90 years. In this survey, I present the history of the problem, and discuss some of the ideas used in the recent breakthrough of Campos–Griffiths–Morris–Sahasrabudhe, who proved that N=3.993^k is sufficient. In addition, I discuss the subsequent work of Balister, Bollobás, Campos, Griffiths, Hurley, Morris, Sahasrabudhe, and Tiba, who gave an alternative, and more conceptual, proof.}, author = {Wigderson, Yuval}, issn = {0303-1179}, journal = {Astérisque}, pages = {85--138}, publisher = {Societe Mathematique de France}, title = {{Exposé Bourbaki 1230 : Upper bounds on diagonal Ramsey numbers (after Campos, Griffiths, Morris, and Sahasrabudhe)}}, doi = {10.24033/ast.1255}, year = {2026}, } @article{22218, abstract = {Physiological void spaces exist at every scale of the human body, from organs to molecules, facilitating transport, signal propagation, and localized biochemical activity. Constriction of these spaces (e.g., arterial occlusion, fibrosis) highlights their importance, making their mimicry essential in tissue engineering (TE). This review examines four key strategies for introducing porosity into hydrogels across multiple length scales: templating, microgels, phase separation, and 3D printing. The first three methods enable the engineering of physiological environments at the nano‐ to micro‐scale, mimicking tissue‐ and extracellular matrix (ECM)‐level spaces. Templating involves embedding and removal of gas, liquid, or solid phases, leaving behind pores. Microgel annealing generates inherent interstitial voids. Liquid–liquid phase separation (LLPS) creates biphasic networks reminiscent of native ECM. The fourth approach, extrusion‐ and light‐based 3D printing techniques, enables the fabrication of larger‐scale spaces, such as luminal structures (e.g., vasculature, airways, and ducts). Combining these methods enables the creation of hierarchical architectures from the nano‐ to centimeter scale. The review also highlights Filamented Light (FLight) technology, which creates internal microstructural voids relevant to anisotropic tissues. This review offers insights into current methods and their convergence for generating biomimetic void spaces to meet the physiological demands of cells, tissues, and organs.}, author = {Puiggalí‐Jou, Anna and Hui, Isabel B. and Fernández-Rico, Carla and Zenobi‐Wong, Marcy}, issn = {1521-4095}, journal = {Advanced Materials}, number = {7}, publisher = {Wiley}, title = {{The space within: How architected voids promote tissue formation}}, doi = {10.1002/adma.202507385}, volume = {38}, year = {2026}, } @article{22215, abstract = {Elastic MicroPhase separation (EMPS) provides a simple route to create soft materials with homogeneous microstructures by leveraging the supersaturation of crosslinked polymer networks with liquids. At low supersaturation, network elasticity stabilizes a uniform mixture, but beyond a critical threshold, metastable microphase-separated domains emerge. While previous theories have focused on describing qualitative features about the size and morphology of these domains, they do not make quantitative predictions about EMPS phase diagrams. In this work, we extend Flory–Huggins theory to quantitatively capture EMPS phase diagrams by incorporating strain-stiffening effects. This model requires no fitting parameters and relies solely on independently measured solubility parameters and large-deformation mechanical responses. Our results confirm that strain-stiffening enables metastable microphase separation within the swelling equilibrium state and reveal why the microstructures can range from discrete droplets to bicontinuous networks. This works highlights the critical role of nonlinear elasticity in controlling phase-separated morphologies in polymer gels.}, author = {Fernández-Rico, Carla and Style, Robert W. and Heyden, Stefanie and Wang, Shichen and Olmsted, Peter D. and Dufresne, Eric R.}, issn = {1744-6848}, journal = {Soft Matter}, number = {2}, pages = {330--342}, publisher = {Royal Society of Chemistry}, title = {{Thermodynamics of microphase separation in a swollen, strain-stiffening polymer network}}, doi = {10.1039/d5sm00594a}, volume = {22}, year = {2026}, } @article{21002, abstract = {The Davenport–Heilbronn method is a version of the circle method that was developed for studying Diophantine inequalities in the paper (Davenport and Heilbronn, J. Lond. Math. Soc. (1) 21 (1946), 185–193). We discuss the main ideas in the paper, together with an account of the development of the subject in the intervening 80 years.}, author = {Browning, Timothy D}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {1}, publisher = {Wiley}, title = {{The Davenport–Heilbronn method: 80 years on}}, doi = {10.1112/jlms.70371}, volume = {113}, year = {2026}, } @article{21385, abstract = {We prove that the average size of a mixed character sum (math. formular) (for a suitable smooth function w) is on the order of √x for all irrational real θ satisfying a weak Diophantine condition, where χ is drawn from the family of Dirichlet characters modulo a large prime r and where x 6 r. In contrast, it was proved by Harper that the average size is o(√x) for rational θ. Certain quadratic Diophantine equations play a key role in the present paper. }, author = {Wang, Victor and Xu, Max}, issn = {1473-7124}, journal = {Proceedings of the Royal Society of Edinburgh: Section A Mathematics}, pages = {1--15}, publisher = {Cambridge University Press}, title = {{Average sizes of mixed character sums}}, doi = {10.1017/prm.2026.10123}, year = {2026}, } @article{21242, abstract = {We obtain an asymptotic formula for the number of integral solutions to a system of diagonal equations. We obtain an asymptotic formula for the number of solutions with variables restricted to smooth numbers as well. We improve the required number of variables compared to previous results by incorporating recent progress on Waring’s problem and the resolution of the main conjecture in Vinogradov’s mean value theorem.}, author = {Rome, Nick and Yamagishi, Shuntaro}, issn = {1945-5844}, journal = {Pacific Journal of Mathematics}, number = {1}, pages = {179--198}, publisher = {Mathematical Sciences Publishers}, title = {{Integral solutions to systems of diagonal equations}}, doi = {10.2140/pjm.2026.340.179}, volume = {340}, year = {2026}, } @article{22318, abstract = {Many intended uses of differential privacy involve a continual mechanism that is set up to run continuously over a long period of time, making more statistical releases as either queries come in or the dataset is updated. In this paper, we give the first general treatment of privacy against adaptive adversaries for mechanisms that support dataset updates and a variety of queries, all arbitrarily interleaved. It also models a very general notion of neighboring, that includes both event-level and user-level privacy. We prove several concurrent composition theorems for continual mechanisms, which ensure privacy even when an adversary can interleave its queries and dataset updates to the different composed mechanisms. Previous concurrent composition theorems for differential privacy were only for the case when the dataset is static, with no adaptive updates. We also give the first interactive and continual generalizations of the “parallel composition theorem” for noninteractive differential privacy. Specifically, we show that the analogue of the noninteractive parallel composition theorem holds if either there are no adaptive dataset updates or each of the composed mechanisms satisfies pure differential privacy, but it fails to hold for composing approximately differentially private mechanisms with dataset updates. Thus, we prove a tight new composition theorem for this case. In addition, we prove concurrent filter compositions theorems for the scenarios in which the privacy parameters are adaptively chosen. We extend these results to other measures of differential privacy, including Rényi DP and 𝑓 -DP. We then formalize a set of general conditions on a continual mechanism M that runs multiple continual submechanisms such that the privacy guarantees of M follow directly using the above concurrent composition theorems on the sub-mechanisms, without further privacy loss. This enables us to give a simpler and modular privacy analysis of a recent continual histogram mechanism of Henzinger, Sricharan, and Steiner. In the case of approximate DP, ours is the first proof that shows that its privacy holds against adaptive adversaries. We also provide a framework that simplifies the analysis of local differential privacy when the protocol includes multi-round server-user interactions. Using this result, we simplify the privacy analysis of the core decomposition protocol of Dhulipala, Henzinger, Li, Liu, Sricharan, and Zhu [5].}, author = {Henzinger, Monika H and Safavi Hemami, Roodabeh and Vadhan, Salil}, issn = {2836-6573}, journal = {Proceedings of the ACM on Management of Data}, keywords = {differential privacy, concurrent composition, continual release, continual observation, data streaming, continual mechanisms, concurrent parallel composition, concurrent filter composition}, number = {2}, pages = {1--26}, publisher = {Association for Computing Machinery}, title = {{Concurrent composition for differentially private continual mechanisms}}, doi = {10.1145/3801895}, volume = {4}, year = {2026}, } @inproceedings{22294, abstract = {Modern computer systems store vast amounts of personal data, enabling advances in AI and ML but risking user privacy and trust. For privacy reasons, it is sometimes desired for an ML model to forget part of the data it was trained on. In this paper, we introduce a novel unlearning approach based on Forgetting Neural Networks (FNNs), a neuroscience-inspired architecture that explicitly encodes forgetting through multiplicative decay factors. While FNNs had previously been studied as a theoretical construct, we provide the first concrete implementation and demonstrate their effectiveness for targeted unlearning. We propose several variants with per-neuron forgetting factors, including rank-based assignments guided by activation levels, and evaluate them on MNIST and Fashion-MNIST benchmarks. Our method systematically removes information associated with forget sets while preserving performance on retained data. Membership inference attacks confirm the effectiveness of FNN-based unlearning in erasing information about the training data from the neural network. These results establish FNNs as a promising foundation for efficient and interpretable unlearning. }, author = {Hatua, Amartya and Nguyen, Trung and Cano Cordoba, Filip and Sung, Andrew}, booktitle = {Proceedings of the 18th International Conference on Agents and Artificial Intelligence}, isbn = {9789897587962}, issn = {2184-433X}, keywords = {Machine Unlearning, Neuroscience-Inspired Machine Learning, Membership Inference Attacks}, location = {Marbella, Spain}, pages = {1536--1546}, publisher = {SciTePress}, title = {{Machine unlearning using forgetting neural networks}}, doi = {10.5220/0014326500004052}, volume = {2}, year = {2026}, } @article{22322, abstract = {We study the problem of continually releasing statistics of an evolving dataset under differential privacy. In the event-level setting, we show the first polynomial lower bounds on the additive error for insertions-only graph problems such as maximum matching, degree histogram and k-core number computation. These results represent an exponential improvement on the polylogarithmic lower bounds of Fichtenberger, Henzinger and Ost [ESA 2021] for the former two problems, and are the first lower bounds in the continual release setting for the latter problem. Our results run counter to the intuition that the difference between insertions-only vs fully dynamic updates causes the gap between polylogarithmic and polynomial additive error. Indeed, we show that for estimating the size of the maximum matching or k-core number of a vertex, allowing small multiplicative approximations is what brings the additive error down to polylogarithmic. We complement these results with improved upper bounds on the additive error when no multiplicative approximation is allowed. Beyond graphs, our techniques also show that polynomial additive error is unavoidable for the Simultaneous Norm Estimation problem in the insertions-only setting. When multiplicative approximations are allowed, we circumvent this lower bound by giving the first continual mechanism with polylogarithmic additive error under (1 + ζ) multiplicative approximations, for any ζ > 0, for estimating all monotone symmetric norms simultaneously. In the item-level setting, we show polynomial lower bounds on the product of the multiplicative and the additive error of continual mechanisms for a large range of graph problems. To the best of our knowledge, these are the first lower bounds shown for any differentially private mechanism under continual release with multiplicative error. To obtain these results, we prove a new lower bound on the product of multiplicative and additive error for the 1-Way-Marginals problem, and give reductions from 1-Way-Marginals to our desired graph problems. This generalizes the prior results of Hardt and Talwar [STOC 2010] and Bun, Ullman and Vadhan [STOC 2014, SIAM J. Comput. 2018], who gave lower bounds on the additive error for the special case of mechanisms with no multiplicative error.}, author = {Aryanfard, Bardiya and Henzinger, Monika H and Saulpic, David and Sricharan, A. R.}, issn = {2836-6573}, journal = {Proceedings of the ACM on Management of Data}, number = {2}, pages = {1--27}, publisher = {Association for Computing Machinery}, title = {{Improved lower bounds for privacy under continual release}}, doi = {10.1145/3801903}, volume = {4}, year = {2026}, } @article{22326, abstract = {In many developmental systems, cells differentiate into a tissue by reading out morphogen concentration fields, a process fundamentally limited by noise. How much can the precision of this process be improved by nonlocal information, e.g., via cell-cell communication? Using a Bayes-optimal framework, we show that positional inference depends crucially on morphogen spatial correlations and on the "structural prior" that encodes the geometry of the cellular lattice performing the readout, thereby determining what a cell can reliably assume about the position of its neighbors when interpreting nonlocal morphogen signals. We derive upper bounds on positional information gain due to nonlocal readout and identify signal processing algorithms that approximate optimal positional inference, as well as simple chemical reaction schemes which implement such algorithms. Our theory suggests that correlational information can be exploited to significantly enhance developmental precision.}, author = {Zhang, Chen Y and Mateu Hoyos, Pablo and Brückner, David and Tkačik, Gašper}, issn = { 1079-7114}, journal = {Physical Review Letters}, publisher = {American Physical Society}, title = {{Nonlocal decoding of positional and correlational information during development}}, doi = {10.1103/mbjk-v4ym}, volume = {137}, year = {2026}, } @article{22333, abstract = {RNA polymerase II (Pol II) must be assembled in the cytoplasm before it enters the nucleus, where it transcribes protein-coding genes. Although transcription by Pol II is intensively studied, how this central multi-subunit enzyme is made and the role of dedicated assembly factors remains unclear. Here, we report the integrative structural analysis of a native human Pol II from the cytoplasm captured near the end of biogenesis. The complex contains Gdown1 and three biogenesis factors – RPAP2 and the critical small GTPases GPN1 and GPN3. Cryo-EM analysis of the complex reveals how Gdown1 and RPAP2 associate with Pol II and prevent the premature association of transcription factors. Further biochemical and cryo-EM analysis reveals how RPAP2 tethers GPN1–GPN3 to the complex and how the assembly of the RPAP2–GPN1–GPN3 complex is controlled by GTP hydrolysis. The combined results uncover a network of interactions that chaperone cytoplasmic Pol II to prevent aberrant interactions, reveal a molecular switch regulating biogenesis factor association, and suggest a general mechanism for the action of GPN-loop GTPase family of enzymes.}, author = {Hlavata, Annamaria and Neuditschko, Benjamin and Schellhaas, Ulla and Plaschka, Clemens and Herzog, Franz and Bernecky, Carrie A}, issn = {2041-1723}, journal = {Nature Communications}, publisher = {Springer Nature}, title = {{Structure of cytoplasmic RNA polymerase II}}, doi = {10.1038/s41467-026-75416-8}, year = {2026}, } @article{21777, abstract = {The advantageous characteristics attributed to the 19F nucleus have made it a popular target for nuclear magnetic resonance (NMR) once again in recent years. Aside from solution NMR, an increasing number of studies have been conducted applying solid-state magic-angle spinning (MAS) NMR to fluorine-labelled samples. Here, the high chemical shift anisotropy and strong dipolar couplings can be utilised to get structural insights into proteins and measure long distances. Despite increasing popularity and promising benefits, the sensitivity of biomolecular 19F MAS NMR often suffers from slow longitudinal T1 relaxation and therefore long recycle delays. In this work, we expand paramagnetic doping, an approach commonly used to reduce proton T1 relaxation times, to 19F-labelled biological samples. We study the effect of Gd(DTPA) and Gd(DTPA-BMA) on 19F T1 and T2, and 13C T1 and T2 relaxation in a [5-19F13C]-tryptophan-labelled protein via 19F-detected MAS NMR experiments. The observed paramagnetic relaxation enhancement substantially reduces measurement times of 19F MAS NMR experiments without compromising resolution. Additionally, we report the chemical shift assignments of all four fluorotryptophan signals in the 12×39 kDa-large protein TET2 using a mutagenesis approach.}, author = {Becker, Lea Marie and Toscano, Giorgia and Kapitonova, Anna and Singh, Rajkumar and Guillerm, Undina and Lichtenecker, Roman J. and Schanda, Paul}, issn = {2699-0016}, journal = {Magnetic Resonance}, number = {1}, pages = {29--37}, publisher = {Copernicus Publications}, title = {{Accelerated 19F biomolecular magic-angle spinning NMR with paramagnetic dopants}}, doi = {10.5194/mr-7-29-2026}, volume = {7}, year = {2026}, } @article{22360, abstract = {In this paper, we consider the rectangular random matrix X =(xij ) ∈ RN×n whose entries are iid with tail P(|xij | > t) ∼ t−α for some α> 0. We consider the regime N(n)/n → a > 1 as n tends to infinity. Our main interest lies in the right singular vector corresponding to the smallest singular value, which we will refer to as the ``bottom singular vector'', denoted by 𝔲. In this paper, we prove the following phase transition regarding the localization length of 𝔲: when α< 2 the localization length is O(n/ log n); when α> 2 the localization length is of order n. Similar results hold for all right singular vectors around the smallest singular value. The variational definition of the bottom singular vector suggests that the mechanism for this localization-delocalization transition when α goes across 2 is intrinsically different from the one for the top singular vector when α goes across 4}, author = {Bao, Zhigang and Lee, Jaehun and Xu, Xiaocong}, issn = {1096-0783}, journal = {Journal of Functional Analysis}, number = {4}, publisher = {Elsevier}, title = {{Phase transition for the bottom singular vector of rectangular random matrices}}, doi = {10.1016/j.jfa.2025.111266}, volume = {290}, year = {2026}, }