@article{13274, abstract = {Viscous flows through pipes and channels are steady and ordered until, with increasing velocity, the laminar motion catastrophically breaks down and gives way to turbulence. How this apparently discontinuous change from low- to high-dimensional motion can be rationalized within the framework of the Navier-Stokes equations is not well understood. Exploiting geometrical properties of transitional channel flow we trace turbulence to far lower Reynolds numbers (Re) than previously possible and identify the complete path that reversibly links fully turbulent motion to an invariant solution. This precursor of turbulence destabilizes rapidly with Re, and the accompanying explosive increase in attractor dimension effectively marks the transition between deterministic and de facto stochastic dynamics.}, author = {Paranjape, Chaitanya S and Yalniz, Gökhan and Duguet, Yohann and Budanur, Nazmi B and Hof, Björn}, issn = {1079-7114}, journal = {Physical Review Letters}, keywords = {General Physics and Astronomy}, number = {3}, publisher = {American Physical Society}, title = {{Direct path from turbulence to time-periodic solutions}}, doi = {10.1103/physrevlett.131.034002}, volume = {131}, year = {2023}, } @article{12788, abstract = {We show that the simplest of existing molecules—closed-shell diatomics not interacting with one another—host topological charges when driven by periodic far-off-resonant laser pulses. A periodically kicked molecular rotor can be mapped onto a “crystalline” lattice in angular momentum space. This allows us to define quasimomenta and the band structure in the Floquet representation, by analogy with the Bloch waves of solid-state physics. Applying laser pulses spaced by 1/3 of the molecular rotational period creates a lattice with three atoms per unit cell with staggered hopping. Within the synthetic dimension of the laser strength, we discover Dirac cones with topological charges. These Dirac cones, topologically protected by reflection and time-reversal symmetry, are reminiscent of (although not equivalent to) that seen in graphene. They—and the corresponding edge states—are broadly tunable by adjusting the laser strength and can be observed in present-day experiments by measuring molecular alignment and populations of rotational levels. This paves the way to study controllable topological physics in gas-phase experiments with small molecules as well as to classify dynamical molecular states by their topological invariants.}, author = {Karle, Volker and Ghazaryan, Areg and Lemeshko, Mikhail}, issn = {1079-7114}, journal = {Physical Review Letters}, number = {10}, publisher = {American Physical Society}, title = {{Topological charges of periodically kicked molecules}}, doi = {10.1103/PhysRevLett.130.103202}, volume = {130}, year = {2023}, } @article{12787, abstract = {Populations evolve in spatially heterogeneous environments. While a certain trait might bring a fitness advantage in some patch of the environment, a different trait might be advantageous in another patch. Here, we study the Moran birth–death process with two types of individuals in a population stretched across two patches of size N, each patch favouring one of the two types. We show that the long-term fate of such populations crucially depends on the migration rate μ between the patches. To classify the possible fates, we use the distinction between polynomial (short) and exponential (long) timescales. We show that when μ is high then one of the two types fixates on the whole population after a number of steps that is only polynomial in N. By contrast, when μ is low then each type holds majority in the patch where it is favoured for a number of steps that is at least exponential in N. Moreover, we precisely identify the threshold migration rate μ⋆ that separates those two scenarios, thereby exactly delineating the situations that support long-term coexistence of the two types. We also discuss the case of various cycle graphs and we present computer simulations that perfectly match our analytical results.}, author = {Svoboda, Jakub and Tkadlec, Josef and Kaveh, Kamran and Chatterjee, Krishnendu}, issn = {1471-2946}, journal = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}, number = {2271}, publisher = {The Royal Society}, title = {{Coexistence times in the Moran process with environmental heterogeneity}}, doi = {10.1098/rspa.2022.0685}, volume = {479}, year = {2023}, } @unpublished{15039, abstract = {A crucial property for achieving secure, trustworthy and interpretable deep learning systems is their robustness: small changes to a system's inputs should not result in large changes to its outputs. Mathematically, this means one strives for networks with a small Lipschitz constant. Several recent works have focused on how to construct such Lipschitz networks, typically by imposing constraints on the weight matrices. In this work, we study an orthogonal aspect, namely the role of the activation function. We show that commonly used activation functions, such as MaxMin, as well as all piece-wise linear ones with two segments unnecessarily restrict the class of representable functions, even in the simplest one-dimensional setting. We furthermore introduce the new N-activation function that is provably more expressive than currently popular activation functions. We provide code at this https URL.}, author = {Prach, Bernd and Lampert, Christoph}, booktitle = {arXiv}, title = {{1-Lipschitz neural networks are more expressive with N-activations}}, doi = {10.48550/ARXIV.2311.06103}, year = {2023}, } @article{13212, abstract = {Auxin is the major plant hormone regulating growth and development (Friml, 2022). Forward genetic approaches in the model plant Arabidopsis thaliana have identified major components of auxin signalling and established the canonical mechanism mediating transcriptional and thus developmental reprogramming. In this textbook view, TRANSPORT INHIBITOR RESPONSE 1 (TIR1)/AUXIN-SIGNALING F-BOX (AFBs) are auxin receptors, which act as F-box subunits determining the substrate specificity of the Skp1-Cullin1-F box protein (SCF) type E3 ubiquitin ligase complex. Auxin acts as a “molecular glue” increasing the affinity between TIR1/AFBs and the Aux/IAA repressors. Subsequently, Aux/IAAs are ubiquitinated and degraded, thus releasing auxin transcription factors from their repression making them free to mediate transcription of auxin response genes (Yu et al., 2022). Nonetheless, accumulating evidence suggests existence of rapid, non-transcriptional responses downstream of TIR1/AFBs such as auxin-induced cytosolic calcium (Ca2+) transients, plasma membrane depolarization and apoplast alkalinisation, all converging on the process of root growth inhibition and root gravitropism (Li et al., 2022). Particularly, these rapid responses are mostly contributed by predominantly cytosolic AFB1, while the long-term growth responses are mediated by mainly nuclear TIR1 and AFB2-AFB5 (Li et al., 2021; Prigge et al., 2020; Serre et al., 2021). How AFB1 conducts auxin-triggered rapid responses and how it is different from TIR1 and AFB2-AFB5 remains elusive. Here, we compare the roles of TIR1 and AFB1 in transcriptional and rapid responses by modulating their subcellular localization in Arabidopsis and by testing their ability to mediate transcriptional responses when part of the minimal auxin circuit reconstituted in yeast.}, author = {Chen, Huihuang and Li, Lanxin and Zou, Minxia and Qi, Linlin and Friml, Jiří}, issn = {1674-2052}, journal = {Molecular Plant}, number = {7}, pages = {1117--1119}, publisher = {Elsevier }, title = {{Distinct functions of TIR1 and AFB1 receptors in auxin signalling.}}, doi = {10.1016/j.molp.2023.06.007}, volume = {16}, year = {2023}, } @inproceedings{13221, abstract = {The safety-liveness dichotomy is a fundamental concept in formal languages which plays a key role in verification. Recently, this dichotomy has been lifted to quantitative properties, which are arbitrary functions from infinite words to partially-ordered domains. We look into harnessing the dichotomy for the specific classes of quantitative properties expressed by quantitative automata. These automata contain finitely many states and rational-valued transition weights, and their common value functions Inf, Sup, LimInf, LimSup, LimInfAvg, LimSupAvg, and DSum map infinite words into the totallyordered domain of real numbers. In this automata-theoretic setting, we establish a connection between quantitative safety and topological continuity and provide an alternative characterization of quantitative safety and liveness in terms of their boolean counterparts. For all common value functions, we show how the safety closure of a quantitative automaton can be constructed in PTime, and we provide PSpace-complete checks of whether a given quantitative automaton is safe or live, with the exception of LimInfAvg and LimSupAvg automata, for which the safety check is in ExpSpace. Moreover, for deterministic Sup, LimInf, and LimSup automata, we give PTime decompositions into safe and live automata. These decompositions enable the separation of techniques for safety and liveness verification for quantitative specifications.}, author = {Boker, Udi and Henzinger, Thomas A and Mazzocchi, Nicolas Adrien and Sarac, Naci E}, booktitle = {34th International Conference on Concurrency Theory}, isbn = {9783959772990}, issn = {1868-8969}, location = {Antwerp, Belgium}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Safety and liveness of quantitative automata}}, doi = {10.4230/LIPIcs.CONCUR.2023.17}, volume = {279}, year = {2023}, } @inproceedings{13143, abstract = {GIMPS and PrimeGrid are large-scale distributed projects dedicated to searching giant prime numbers, usually of special forms like Mersenne and Proth primes. The numbers in the current search-space are millions of digits large and the participating volunteers need to run resource-consuming primality tests. Once a candidate prime N has been found, the only way for another party to independently verify the primality of N used to be by repeating the expensive primality test. To avoid the need for second recomputation of each primality test, these projects have recently adopted certifying mechanisms that enable efficient verification of performed tests. However, the mechanisms presently in place only detect benign errors and there is no guarantee against adversarial behavior: a malicious volunteer can mislead the project to reject a giant prime as being non-prime. In this paper, we propose a practical, cryptographically-sound mechanism for certifying the non-primality of Proth numbers. That is, a volunteer can – parallel to running the primality test for N – generate an efficiently verifiable proof at a little extra cost certifying that N is not prime. The interactive protocol has statistical soundness and can be made non-interactive using the Fiat-Shamir heuristic. Our approach is based on a cryptographic primitive called Proof of Exponentiation (PoE) which, for a group G, certifies that a tuple (x,y,T)∈G2×N satisfies x2T=y (Pietrzak, ITCS 2019 and Wesolowski, J. Cryptol. 2020). In particular, we show how to adapt Pietrzak’s PoE at a moderate additional cost to make it a cryptographically-sound certificate of non-primality.}, author = {Hoffmann, Charlotte and Hubáček, Pavel and Kamath, Chethan and Pietrzak, Krzysztof Z}, booktitle = {Public-Key Cryptography - PKC 2023}, isbn = {9783031313677}, issn = {1611-3349}, location = {Atlanta, GA, United States}, pages = {530--553}, publisher = {Springer Nature}, title = {{Certifying giant nonprimes}}, doi = {10.1007/978-3-031-31368-4_19}, volume = {13940}, year = {2023}, } @article{13097, abstract = {Vertebrate movement is orchestrated by spinal inter- and motor neurons that, together with sensory and cognitive input, produce dynamic motor behaviors. These behaviors vary from the simple undulatory swimming of fish and larval aquatic species to the highly coordinated running, reaching and grasping of mice, humans and other mammals. This variation raises the fundamental question of how spinal circuits have changed in register with motor behavior. In simple, undulatory fish, exemplified by the lamprey, two broad classes of interneurons shape motor neuron output: ipsilateral-projecting excitatory neurons, and commissural-projecting inhibitory neurons. An additional class of ipsilateral inhibitory neurons is required to generate escape swim behavior in larval zebrafish and tadpoles. In limbed vertebrates, a more complex spinal neuron composition is observed. In this review, we provide evidence that movement elaboration correlates with an increase and specialization of these three basic interneuron types into molecularly, anatomically, and functionally distinct subpopulations. We summarize recent work linking neuron types to movement-pattern generation across fish, amphibians, reptiles, birds and mammals.}, author = {Wilson, Alexia C and Sweeney, Lora Beatrice Jaeger}, issn = {1662-5110}, journal = {Frontiers in Neural Circuits}, publisher = {Frontiers}, title = {{Spinal cords: Symphonies of interneurons across species}}, doi = {10.3389/fncir.2023.1146449}, volume = {17}, year = {2023}, } @article{14421, abstract = {Only recently has it been possible to construct a self-adjoint Hamiltonian that involves the creation of Dirac particles at a point source in 3d space. Its definition makes use of an interior-boundary condition. Here, we develop for this Hamiltonian a corresponding theory of the Bohmian configuration. That is, we (non-rigorously) construct a Markov jump process $(Q_t)_{t\in\mathbb{R}}$ in the configuration space of a variable number of particles that is $|\psi_t|^2$-distributed at every time t and follows Bohmian trajectories between the jumps. The jumps correspond to particle creation or annihilation events and occur either to or from a configuration with a particle located at the source. The process is the natural analog of Bell's jump process, and a central piece in its construction is the determination of the rate of particle creation. The construction requires an analysis of the asymptotic behavior of the Bohmian trajectories near the source. We find that the particle reaches the source with radial speed 0, but orbits around the source infinitely many times in finite time before absorption (or after emission).}, author = {Henheik, Sven Joscha and Tumulka, Roderich}, issn = {1751-8121}, journal = {Journal of Physics A: Mathematical and Theoretical}, number = {44}, publisher = {IOP Publishing}, title = {{Creation rate of Dirac particles at a point source}}, doi = {10.1088/1751-8121/acfe62}, volume = {56}, year = {2023}, } @article{13317, abstract = {We prove the Eigenstate Thermalisation Hypothesis (ETH) for local observables in a typical translation invariant system of quantum spins with L-body interactions, where L is the number of spins. This mathematically verifies the observation first made by Santos and Rigol (Phys Rev E 82(3):031130, 2010, https://doi.org/10.1103/PhysRevE.82.031130) that the ETH may hold for systems with additional translational symmetries for a naturally restricted class of observables. We also present numerical support for the same phenomenon for Hamiltonians with local interaction.}, author = {Sugimoto, Shoki and Henheik, Sven Joscha and Riabov, Volodymyr and Erdös, László}, issn = {1572-9613}, journal = {Journal of Statistical Physics}, number = {7}, publisher = {Springer Nature}, title = {{Eigenstate thermalisation hypothesis for translation invariant spin systems}}, doi = {10.1007/s10955-023-03132-4}, volume = {190}, year = {2023}, } @article{14343, abstract = {The total energy of an eigenstate in a composite quantum system tends to be distributed equally among its constituents. We identify the quantum fluctuation around this equipartition principle in the simplest disordered quantum system consisting of linear combinations of Wigner matrices. As our main ingredient, we prove the Eigenstate Thermalisation Hypothesis and Gaussian fluctuation for general quadratic forms of the bulk eigenvectors of Wigner matrices with an arbitrary deformation.}, author = {Cipolloni, Giorgio and Erdös, László and Henheik, Sven Joscha and Kolupaiev, Oleksii}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Gaussian fluctuations in the equipartition principle for Wigner matrices}}, doi = {10.1017/fms.2023.70}, volume = {11}, year = {2023}, } @inproceedings{17378, abstract = {Generative Pre-trained Transformer models, known as GPT or OPT, set themselves apart through breakthrough performance across complex language modelling tasks, but also by their extremely high computational and storage costs. Specifically, due to their massive size, even inference for large, highly-accurate GPT models may require multiple performant GPUs, which limits the usability of such models. While there is emerging work on relieving this pressure via model compression, the applicability and performance of existing compression techniques is limited by the scale and complexity of GPT models. In this paper, we address this challenge, and propose OPTQ, a new one-shot weight quantization method based on approximate second-order information, that is both highly-accurate and highly-efficient. Specifically, OPTQ can quantize GPT models with 175 billion parameters in approximately four GPU hours, reducing the bitwidth down to 3 or 4 bits per weight, with negligible accuracy degradation relative to the uncompressed baseline. Our method more than doubles the compression gains relative to previously-proposed one-shot quantization methods, preserving accuracy, allowing us for the first time to execute an 175 billion-parameter model inside a single GPU for generative inference. Moreover, we also show that our method can still provide reasonable accuracy in the extreme quantization regime, in which weights are quantized to 2-bit or even ternary quantization levels. We show experimentally that these improvements can be leveraged for end-to-end inference speedups over FP16, of around 3.25x when using high-end GPUs (NVIDIA A100) and 4.5x when using more cost-effective ones (NVIDIA A6000). The implementation is available at https://github.com/IST-DASLab/gptq.}, author = {Frantar, Elias and Ashkboos, Saleh and Hoefler, Torsten and Alistarh, Dan-Adrian}, booktitle = {11th International Conference on Learning Representations }, location = {Kigali, Rwanda}, publisher = {International Conference on Learning Representations}, title = {{OPTQ: Accurate post-training quantization for generative pre-trained transformers}}, year = {2023}, } @inproceedings{14458, abstract = {We show for the first time that large-scale generative pretrained transformer (GPT) family models can be pruned to at least 50% sparsity in one-shot, without any retraining, at minimal loss of accuracy. This is achieved via a new pruning method called SparseGPT, specifically designed to work efficiently and accurately on massive GPT-family models. We can execute SparseGPT on the largest available open-source models, OPT-175B and BLOOM-176B, in under 4.5 hours, and can reach 60% unstructured sparsity with negligible increase in perplexity: remarkably, more than 100 billion weights from these models can be ignored at inference time. SparseGPT generalizes to semi-structured (2:4 and 4:8) patterns, and is compatible with weight quantization approaches. The code is available at: https://github.com/IST-DASLab/sparsegpt.}, author = {Frantar, Elias and Alistarh, Dan-Adrian}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, issn = {2640-3498}, location = {Honolulu, Hawaii, HI, United States}, pages = {10323--10337}, publisher = {ML Research Press}, title = {{SparseGPT: Massive language models can be accurately pruned in one-shot}}, volume = {202}, year = {2023}, } @unpublished{18294, abstract = {Using a two-dimensional version of the delta method, we establish an asymptotic formula for the number of rational points of bounded height on non-singular complete intersections of cubic and quadric hypersurfaces of dimension at least 23 over Fq(t), provided cha(Fq)>3. Under the same hypotheses, we also verify weak approximation.}, author = {Glas, Jakob}, booktitle = {arXiv}, title = {{Complete intersections of cubic and quadric hypersurfaces over Fq(t)}}, doi = {10.48550/arXiv.2306.02718}, year = {2023}, } @unpublished{14732, abstract = {Fragmented landscapes pose a significant threat to the persistence of species as they are highly susceptible to heightened risk of extinction due to the combined effects of genetic and demographic factors such as genetic drift and demographic stochasticity. This paper explores the intricate interplay between genetic load and extinction risk within metapopulations with a focus on understanding the impact of eco-evolutionary feedback mechanisms. We distinguish between two models of selection: soft selection, characterised by subpopulations maintaining carrying capacity despite load, and hard selection, where load can significantly affect population size. Within the soft selection framework, we investigate the impact of gene flow on genetic load at a single locus, while also considering the effect of selection strength and dominance coefficient. We subsequently build on this to examine how gene flow influences both population size and load under hard selection as well as identify critical thresholds for metapopulation persistence. Our analysis employs the diffusion, semi-deterministic and effective migration approximations. Our findings reveal that under soft selection, even modest levels of migration can significantly alleviate the burden of load. In sharp contrast, with hard selection, a much higher degree of gene flow is required to mitigate load and prevent the collapse of the metapopulation. Overall, this study sheds light into the crucial role migration plays in shaping the dynamics of genetic load and extinction risk in fragmented landscapes, offering valuable insights for conservation strategies and the preservation of diversity in a changing world.}, author = {Olusanya, Oluwafunmilola O and Khudiakova, Kseniia and Sachdeva, Himani}, booktitle = {bioRxiv}, title = {{Genetic load, eco-evolutionary feedback and extinction in a metapopulation}}, doi = {10.1101/2023.12.02.569702}, year = {2023}, } @article{12334, abstract = {Regulation of the Arp2/3 complex is required for productive nucleation of branched actin networks. An emerging aspect of regulation is the incorporation of subunit isoforms into the Arp2/3 complex. Specifically, both ArpC5 subunit isoforms, ArpC5 and ArpC5L, have been reported to fine-tune nucleation activity and branch junction stability. We have combined reverse genetics and cellular structural biology to describe how ArpC5 and ArpC5L differentially affect cell migration. Both define the structural stability of ArpC1 in branch junctions and, in turn, by determining protrusion characteristics, affect protein dynamics and actin network ultrastructure. ArpC5 isoforms also affect the positioning of members of the Ena/Vasodilator-stimulated phosphoprotein (VASP) family of actin filament elongators, which mediate ArpC5 isoform–specific effects on the actin assembly level. Our results suggest that ArpC5 and Ena/VASP proteins are part of a signaling pathway enhancing cell migration.}, author = {Fäßler, Florian and Javoor, Manjunath and Datler, Julia and Döring, Hermann and Hofer, Florian and Dimchev, Georgi A and Hodirnau, Victor-Valentin and Faix, Jan and Rottner, Klemens and Schur, Florian KM}, issn = {2375-2548}, journal = {Science Advances}, keywords = {Multidisciplinary}, number = {3}, publisher = {American Association for the Advancement of Science}, title = {{ArpC5 isoforms regulate Arp2/3 complex–dependent protrusion through differential Ena/VASP positioning}}, doi = {10.1126/sciadv.add6495}, volume = {9}, year = {2023}, } @unpublished{17351, abstract = {Contractive coupling rates have been recently introduced by Conforti as a tool to establish convex Sobolev inequalities (including modified log-Sobolev and Poincar\'{e} inequality) for some classes of Markov chains. In this work, we show how contractive coupling rates can also be used to prove stronger inequalities, in the form of curvature lower bounds for Markov chains and geodesic convexity of entropic functionals. We illustrate this in several examples discussed by Conforti, where in particular, after appropriately choosing a parameter function, we establish positive curvature in the entropic and (discrete) Bakry--\'{E}mery sense. In addition, we recall and give straightforward generalizations of some notions of coarse Ricci curvature, and we discuss some of their properties and relations with the concepts of couplings and coupling rates: as an application, we show exponential contraction of the $p$-Wasserstein distance for the heat flow in the aforementioned examples.}, author = {Pedrotti, Francesco}, booktitle = {arXiv}, title = {{Contractive coupling rates and curvature lower bounds for Markov chains}}, doi = {10.48550/arXiv.2308.00516}, year = {2023}, } @inproceedings{14461, abstract = {Communication-reduction techniques are a popular way to improve scalability in data-parallel training of deep neural networks (DNNs). The recent emergence of large language models such as GPT has created the need for new approaches to exploit data-parallelism. Among these, fully-sharded data parallel (FSDP) training is highly popular, yet it still encounters scalability bottlenecks. One reason is that applying compression techniques to FSDP is challenging: as the vast majority of the communication involves the model’s weights, direct compression alters convergence and leads to accuracy loss. We present QSDP, a variant of FSDP which supports both gradient and weight quantization with theoretical guarantees, is simple to implement and has essentially no overheads. To derive QSDP we prove that a natural modification of SGD achieves convergence even when we only maintain quantized weights, and thus the domain over which we train consists of quantized points and is, therefore, highly non-convex. We validate this approach by training GPT-family models with up to 1.3 billion parameters on a multi-node cluster. Experiments show that QSDP preserves model accuracy, while completely removing the communication bottlenecks of FSDP, providing end-to-end speedups of up to 2.2x.}, author = {Markov, Ilia and Vladu, Adrian and Guo, Qi and Alistarh, Dan-Adrian}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, issn = {2640-3498}, location = {Honolulu, Hawaii, HI, United States}, pages = {24020--24044}, publisher = {ML Research Press}, title = {{Quantized distributed training of large models with convergence guarantees}}, volume = {202}, year = {2023}, } @unpublished{17174, abstract = {We prove that a class of weakly perturbed Hamiltonians of the form $H_λ= H_0 + λW$, with $W$ being a Wigner matrix, exhibits prethermalization. That is, the time evolution generated by $H_λ$ relaxes to its ultimate thermal state via an intermediate prethermal state with a lifetime of order $λ^{-2}$. Moreover, we obtain a general relaxation formula, expressing the perturbed dynamics via the unperturbed dynamics and the ultimate thermal state. The proof relies on a two-resolvent law for the deformed Wigner matrix $H_λ$.}, author = {Erdös, László and Henheik, Sven Joscha and Reker, Jana and Riabov, Volodymyr}, booktitle = {arXiv}, title = {{Prethermalization for deformed Wigner Matrices}}, doi = {10.48550/arXiv.2310.06677}, year = {2023}, } @unpublished{17173, abstract = {Consider the random variable $\mathrm{Tr}( f_1(W)A_1\dots f_k(W)A_k)$ where $W$ is an $N\times N$ Hermitian Wigner matrix, $k\in\mathbb{N}$, and choose (possibly $N$-dependent) regular functions $f_1,\dots, f_k$ as well as bounded deterministic matrices $A_1,\dots,A_k$. We give a functional central limit theorem showing that the fluctuations around the expectation are Gaussian. Moreover, we determine the limiting covariance structure and give explicit error bounds in terms of the scaling of $f_1,\dots,f_k$ and the number of traceless matrices among $A_1,\dots,A_k$, thus extending the results of [Cipolloni, Erdős, Schröder 2023] to products of arbitrary length $k\geq2$. As an application, we consider the fluctuation of $\mathrm{Tr}(\mathrm{e}^{\mathrm{i} tW}A_1\mathrm{e}^{-\mathrm{i} tW}A_2)$ around its thermal value $\mathrm{Tr}(A_1)\mathrm{Tr}(A_2)$ when $t$ is large and give an explicit formula for the variance.}, author = {Reker, Jana}, booktitle = {arXiv}, title = {{Multi-point functional central limit theorem for Wigner Matrices}}, doi = {10.48550/arXiv.2307.11028}, year = {2023}, }