Invited 09

## Quantum Statistics (Organizer: Cristina Butucea)

Conference
9:30 PM — 10:00 PM KST
Local
Jul 22 Thu, 8:30 AM — 9:00 AM EDT

### Estimation of quantum state and quantum channel

Masahito Hayashi (Southern University of Science and Technology)

5
We review the results of quantum state estimation by using Cramer-Rao type bound. The quantum channel estimation has two scaling, standard quantum limit and Heisenberg scaling. In the standard quantum limit, we present that the extension of the above method works well. For the Heisenberg scaling, we present that the method of Fourier transform method works.

### Information geometry and local asymptotic normality for quantum Markov processes

4
This talk deals with the problem of identifying and estimating dynamical parameters of quantum Markov processes, in the input-output formalism. I will discuss several aspects of this problem: The first aspect concerns the structure of the space of identifiable parameters for ergodic dynamics, assuming full access to the output state for arbitrarily long times. I will show that the equivalence classes of undistinguishable parameters are orbits of a Lie group acting on the space of dynamical parameters. The second aspect concerns the information geometric structure on this space. I will show that the space of identifiable parameters and carries a Riemannian metric based on the quantum Fisher information of the output. The metric can be computed explicitly in terms of the Markov covariance of certain fluctuation operators. The third aspect concerns the asymptotic statistical structure of the output state. I will show that the output state satisfy local asymptotic normality, i.e. they can be approximated by a Gaussian model constructed from the Markov covariance data.

### Optimal adaptive strategies for sequential quantum hypothesis testing

Marco Tomamichel (National University of Singapore)

4
We consider sequential hypothesis testing between two quantum states using adaptive and non-adaptive strategies. In this setting, samples of an unknown state are requested sequentially and a decision to either continue or to accept one of the two hypotheses is made after each test. Under the constraint that the number of samples is bounded, either in expectation or with high probability, we exhibit adaptive strategies that minimize both types of misidentification errors. Namely, we show that these errors decrease exponentially (in the stopping time) with decay rates given by the measured relative entropies between the two states. Moreover, if we allow joint measurements on multiple samples, the rates are increased to the respective quantum relative entropies. We also fully characterize the achievable error exponents for non-adaptive strategies and provide numerical evidence showing that adaptive measurements are necessary to achieve our bounds under some additional assumptions.

### Q&A for Invited Session 09

0
This talk does not have an abstract.

###### Session Chair

Cristina Butucea (École nationale de la statistique et de l'administration économique Paris)

Invited 22

## Random Trees (Organizer: Anita Winter)

Conference
9:30 PM — 10:00 PM KST
Local
Jul 22 Thu, 8:30 AM — 9:00 AM EDT

### 1d Brownian loop soup, Fleming-Viot processes and Bass-Burdzy flow

Elie Aidekon (Fudan University)

5
We describe the connection between these three objects which appear in the problem of conditioning the so-called perturbed reflecting Brownian motion on its occupation field.

Joint work with Yueyun Hu and Zhan Shi.

### A new state space of algebraic measure trees for stochastic processes

Wolfgang Löhr (University of Duisburg-Essen)

5
In the talk, I present a new topological space of “continuum” trees, which extends the set of finite graph-theoretic trees to uncountable structures, which can be seen as limits of finite trees. Unlike previous approaches, we do not use the graph-metric but formalize the tree-structure by a tertiary operation on the tree, namely the branch-point map. The resulting space of algebraic measure trees has coarser equivalence classes than the older space of metric measure trees, but the topology preserves more of the tree-structure in limits, so that it is incomparable to, and not coarser than, the standard topologies on metric measure trees. With the example of the Aldous chain on cladograms, I also illustrate that our new space can be very useful as state-space for stochastic processes in order to obtain path-space diffusion limits of tree-valued Markov chains.

### Scaling Limits of critical rank-1 inhomogeneous random graphs

Minmin Wang (University of Sussex)

5
Continuum inhomogeneous random graphs arise in the scaling limits of critical rank-1 inhomogeneous random graphs. They are extensions of the continuum random graph introduced by Addario-Berry, Broutin & Goldschmidt which has appeared as the scaling limit of the Erdos-Renyi graph in the critical window. In this talk, we present a construction of these graphs from the Levy processes without replacement of Aldous & Limic. In particular, this construction reveals a close connection between the clusters of the graphs and Levy trees, which consists in an isometric embedding of the spanning trees of these clusters into Levy trees. As a consequence of this construction, we provide near optimal conditions for the convergence of critical rank-1 inhomogeneous random graphs. We also deduce the Hausdorff dimension and the packing dimension of the limit graphs.

### Q&A for Invited Session 22

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This talk does not have an abstract.

###### Session Chair

Anita Winter (University of Duisburg-Essen)

Invited 29

## High Dimensional Data Inference (Organizer: Florentina Bunea)

Conference
9:30 PM — 10:00 PM KST
Local
Jul 22 Thu, 8:30 AM — 9:00 AM EDT

### Minimax rates for derivative-free stochastic optimization with higher order smooth objectives

Alexandre Tsybakov (Center for Research in Economics and Statistics (CREST))

3
We study the problem of finding the minimizer of a function by a sequential exploration of its values, under measurement noise. In the stochastic optimization literature, this problem is known under the name of derivative-free or zero-order optimization. We consider an approximation of the gradient descent algorithm where the gradient is estimated by procedures involving function evaluations in randomized points and a smoothing kernel. We first assume that the objective function is $\beta$-Hölder and $\alpha$-strongly convex with some $\alpha>0$, and $\beta$ greater or equal to 2. Under general adversarial noise, we obtain non-asymptotic upper bounds, both for the optimization error and the cumulative regret of the algorithm, as functions of the quadruplet $(T, \alpha, \beta, d)$, where T is the number of queries and d is the dimension of the problem. Furthermore, we establish minimax lower bounds for any sequential search method implying that the suggested algorithm is nearly optimal in a minimax sense in terms of sample complexity and the problem parameters. Based on similar ideas, we solve several other problems. In particular, we propose an algorithm achieving almost sharp oracle behavior for the problem of estimating the minimum value of the function. Next, we extend the results to zero-order distributed optimization, where the aim is to minimize the average of local objectives associated to different nodes in a graph with an exchange of information permitted only between neighboring nodes. Finally, we apply similar ideas for the problem of non-convex optimization.

The talk is based on a joint work with Arya Akhavan and Massimiliano Pontil.

### High-dimensional, multiscale online changepoint detection

Richard Samworth (University of Cambridge)

4
We introduce a new method for high-dimensional, online changepoint detection in settings where a $p$-variate Gaussian data stream may undergo a change in mean. The procedure works by performing likelihood ratio tests against simple alternatives of different scales in each coordinate, and then aggregating test statistics across scales and coordinates. The algorithm is online in the sense that both its storage requirements and worst-case computational complexity per new observation are independent of the number of previous observations. We prove that the patience, or average run length under the null, of our procedure is at least at the desired nominal level, and provide guarantees on its response delay under the alternative that depend on the sparsity of the vector of mean change. Numerical results confirm the practical effectiveness of our proposal, which is implemented in the R package 'ocd'.

### Optimal transport and inference for stationary processes

Andrew Nobel (The University of North Carolina at Chapel Hill)

3
Optimal transport ideas have found widespread use in a variety of practical and theoretical statistical problems. In most of these problems the objects under study are fixed and do not possess dynamic structure. However, there are many problems in which the objects of interest are themselves stationary processes, and in these cases it is natural to consider couplings that preserve stationarity. In the talk I will describe several ways in which stationary couplings arise in inference problems for families of processes, how in the Markov setting consideration of transition couplings can lead to fast algorithms for finding optimal couplings, and how optimal couplings may be estimated from data. I will illustrate the potential utility of the Markov case by considering the problem of graph alignment.

### Q&A for Invited Session 29

0
This talk does not have an abstract.

###### Session Chair

Florentina Bunea (Cornell University)

Invited 34

## Random Walks on Random Media (Organizer: Alexander Drewitz)

Conference
9:30 PM — 10:00 PM KST
Local
Jul 22 Thu, 8:30 AM — 9:00 AM EDT

### Random walk on a barely supercritical branching random walk

Jan Nagel (Technische Universität Dortmund)

5
The motivating question behind this project is how a random walk behaves on a barely supercritical percolation cluster, that is, an infinite percolation cluster when the percolation probability is close to the critical value. As a more tractable model, we approximate the percolation cluster by the embedding of a Galton-Watson tree into the lattice. When the random walk runs on the tree, the embedded process is a random walk on a branching random walk. Now we can consider a barely supercritical branching process conditioned on survival, with survival probability approaching zero. In this setting the tree structure allows a fine analysis of the random walk and we can prove a scaling limit for the embedded process under a nonstandard scaling, when the tree becomes more critical the longer the random walk runs on it. This scaling limit allows us to interpolate between supercritical and critical behavior.

### Universality of cutoff for graphs with an added random matching

Perla Sousi (Cambridge University)

4
We establish universality of cutoff for simple random walk on a class of random graphs defined as follows. Given a finite graph G=(V,E) with |V| even we define a random graph $G^{\prime} = (V, E \cup E^{\prime})$ obtained by picking E$^{\prime}$ to be the (unordered) pairs of a random perfect matching of V. We show that for a sequence of such graphs of diverging sizes and of uniformly bounded degree, if the minimal size of a connected component of (the original graphs) is at least 3, then the obtained graphs are w.h.p. expanders and the random walk on them exhibits cutoff at a time with an entropic interpretation. This provides a simple generic operation of adding some randomness to a given graph, which results in cutoff. The emerging "cutoff at an entropic time" paradigm will be emphasized.

### Invariance principle for a random walk among a Poisson field of moving traps

Rongfeng Sun (National University of Singapore)

4
For a random walk among an i.i.d. Bernoulli field of immobile traps on $Z^d$, it is a classic result that conditioned on survival up to time n, the random walk is confined in a ball of radius $R n^{1/(d+2)}$, and the rescaled path converges to a Brownian motion conditioned to stay inside a ball of radius R. When the traps are mobile and perform independent random walks, the only path result so far is that in dimension 1, the random walk (conditioned on survival) is sub-diffusive. We show that in dimension 5 and higher, instead of being confined on a sub-diffusive scale as in the case of immobile traps, the conditioned walk satisfies an invariance principle on the diffusive scale. Our proof is based on the theory of Thermodynamic Formalism.

Joint work with S. Athreya and A. Drewitz.

### Q&A for Invited Session 34

0
This talk does not have an abstract.

###### Session Chair

Alexander Drewitz (Universität zu Köln)

Invited 37

## Bernoulli Society New Researcher Award Session (Organizer: Bernoulli Society)

Conference
9:30 PM — 10:00 PM KST
Local
Jul 22 Thu, 8:30 AM — 9:00 AM EDT

### Hydrodynamic large deviations of strongly asymmetric interacting particle systems

Li-Cheng Tsai (Rutgers University)

4
We consider the large deviations from the hydrodynamic limit of the Totally Asymmetric Simple Exclusion Process (TASEP), which is related to the entropy production in the inviscid Burgers equation. Here we prove the full large deviation principle. Our method relies on the explicit formula of Matetski, Quastel, and Remenik (2016) for the transition probabilities of the TASEP.

### Conformal loop ensembles on Liouville quantum gravity with marked points

Nina Holden (Swiss Federal Institute of Technology Zürich)

3
Liouville quantum gravity (LQG) surfaces are a natural family of random fractal surfaces, while the conformal loop ensemble (CLE) is a random collection of conformally invariant non-crossing loops in the plane. In a joint work with Matthis Lehmkuehler we study CLE-decorated LQG surfaces whose law has been reweighted according to CLE loop nesting statistics around a fixed number of marked points.

### Integrability of Schramm-Loewner evolution and Liouville quantum gravity

Xin Sun (University of Pennsylvania)

2
It appears that there are rich integrable structures in Schramm-Loewner evolution and Liouville quantum gravity. Namely, many important observables admit exact expressions. In this talk, I will review two major resources of such integrability: conformal field theory and random planar maps decorated with statistical physics models. I will then present a recent work with Morris Ang that proves an integrable result for conformal loop ensembles, which is analogous to the DOZZ formula in Liouville conformal field theory. Our result is an example of a series of results that are proved by blending these two sources of integrability.

### Q&A for Invited Session 37

0
This talk does not have an abstract.

###### Session Chair

Imma Curato (Ulm University)