I will describe the application of tensor networks methods for the study of fluctuations in systems with stochastic dynamics. I will discuss several related problems: (i) how to obtain the long-time statistics of trajectory observables from tilted Markov generators (in analogy with finding quantum ground states); (ii) how to exploit these methods to efficiently sample rare events; (iii) how to extend this approach to finite times (in analogy with calculating finite-temperature quantum partition sums); (iv) the emergence of symmetry protected topological phases in dynamical LDs.
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