Exponential concentration of cover times Alex Zhai (azhai@stanford.edu) May 17, 2015 Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 1 / 27
Outline Part I: Preliminaries Effective resistance and Gaussian free fields Ray-Knight theorems Part II: Application to cover times Part III: Stochastic domination in the generalized 2nd Ray-Knight theorem Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 2 / 27
Part I: Preliminaries Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 3 / 27
Our setting G = ( V , E ) a simple graph, and fix a starting vertex v 0 ∈ V . Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 4 / 27
Our setting G = ( V , E ) a simple graph, and fix a starting vertex v 0 ∈ V . We consider continuous time random walks X = { X t } t ∈ R + started at v 0 : same as usual simple random walk, except time between jumps is a standard exponential random variable X t denotes the vertex you’re on at time t Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 4 / 27
Our setting G = ( V , E ) a simple graph, and fix a starting vertex v 0 ∈ V . We consider continuous time random walks X = { X t } t ∈ R + started at v 0 : same as usual simple random walk, except time between jumps is a standard exponential random variable X t denotes the vertex you’re on at time t Define cover time τ cov = the first time all vertices are visited at least once hitting time τ hit ( x , y ) = the first time walk started at x visits y Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 4 / 27
Effective resistance For any x , y ∈ V , imagine all the edges are unit resistors and we connect the ends of a battery to x and y . Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 5 / 27
Effective resistance For any x , y ∈ V , imagine all the edges are unit resistors and we connect the ends of a battery to x and y . Then, define R eff ( x , y ) = effective resistance between x and y Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 5 / 27
Effective resistance For any x , y ∈ V , imagine all the edges are unit resistors and we connect the ends of a battery to x and y . Then, define R eff ( x , y ) = effective resistance between x and y We can compute R eff ( x , y ) by solving for a function f : V → R such that 1 if z = x ∆ f ( z ) = − 1 if z = y 0 otherwise Then R eff ( x , y ) = f ( y ) − f ( x ). Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 5 / 27
Effective resistance For any x , y ∈ V , imagine all the edges are unit resistors and we connect the ends of a battery to x and y . Then, define R eff ( x , y ) = effective resistance between x and y We can compute R eff ( x , y ) by solving for a function f : V → R such that 1 if z = x ∆ f ( z ) = − 1 if z = y 0 otherwise Then R eff ( x , y ) = f ( y ) − f ( x ). Commute time identity: E τ hit ( x , y ) + E τ hit ( y , x ) = | E | · R eff ( x , y ) . 2 Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 5 / 27
Gaussian free field: definition Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 6 / 27
Gaussian free field: definition For a graph G = ( V , E ), the Gaussian free field (GFF) η is a multivariate Gaussian: coordinates η v indexed by v ∈ V , with η v 0 = 0 Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 6 / 27
Gaussian free field: definition For a graph G = ( V , E ), the Gaussian free field (GFF) η is a multivariate Gaussian: coordinates η v indexed by v ∈ V , with η v 0 = 0 for f ∈ R V with f v 0 = 0, − 1 � ( f x − f y ) 2 [probability of f ] ∝ exp 2 ( x , y ) ∈ E Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 6 / 27
Gaussian free field: definition For a graph G = ( V , E ), the Gaussian free field (GFF) η is a multivariate Gaussian: coordinates η v indexed by v ∈ V , with η v 0 = 0 for f ∈ R V with f v 0 = 0, − 1 � ( f x − f y ) 2 [probability of f ] ∝ exp 2 ( x , y ) ∈ E equivalently, E ( η x − η y ) 2 = R eff ( x , y ) (note: E η 2 x = R eff ( x , v 0 )) Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 6 / 27
Gaussian free field: example Below is a realization of the GFF on a discrete 2D lattice: Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 7 / 27
Gaussian free field: example Let { B t } t ≥ 0 be a Brownian motion. GFF of a path is η = (0 = B 0 , B 1 , . . . , B n ) . Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 8 / 27
Local times Reminder: G = ( V , E ) a graph and X t a continuous time random walk. Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 9 / 27
Local times Reminder: G = ( V , E ) a graph and X t a continuous time random walk. For x ∈ V and s ∈ R + , define local time � s 1 1 ( X s ′ = x ) ds ′ L s ( x ) = deg( x ) 0 1 = deg( x ) (time spent by r.w. at x up to time s ) . Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 9 / 27
Return times For any t > 0, define τ + ( t ) = inf { s ≥ 0 : L s ( v 0 ) ≥ t } = first time that v 0 accumulates local time t . Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 10 / 27
Return times For any t > 0, define τ + ( t ) = inf { s ≥ 0 : L s ( v 0 ) ≥ t } = first time that v 0 accumulates local time t . Remark: τ + � � 1 is like the return time of a discrete time deg( v 0 ) random walk. Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 10 / 27
Return times For any t > 0, define τ + ( t ) = inf { s ≥ 0 : L s ( v 0 ) ≥ t } = first time that v 0 accumulates local time t . Remark: τ + � � 1 is like the return time of a discrete time deg( v 0 ) random walk. We have E τ + ( t ) = 2 | E | · t . (Analogous to expected return time being equal to inverse stationary probability.) Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 10 / 27
Generalized 2nd Ray-Knight theorem Theorem (Generalized Second Ray-Knight Theorem) Let X be a continuous time random walk, and let η and η ′ be GFFs with X and η independent. Then, for any t > 0 , � � � 1 � 2 � √ L τ + ( t ) ( x ) + 1 � law 2 η 2 η ′ = x + 2 t . x 2 x ∈ V x ∈ V Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 11 / 27
Generalized 2nd Ray-Knight theorem Theorem (Generalized Second Ray-Knight Theorem) Let X be a continuous time random walk, and let η and η ′ be GFFs with X and η independent. Then, for any t > 0 , � � � 1 � 2 � √ L τ + ( t ) ( x ) + 1 � law 2 η 2 η ′ = x + 2 t . x 2 x ∈ V x ∈ V Above theorem due to Eisenbaum-Kaspi-Marcus-Rosen-Shi. Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 11 / 27
Generalized 2nd Ray-Knight theorem Theorem (Generalized Second Ray-Knight Theorem) Let X be a continuous time random walk, and let η and η ′ be GFFs with X and η independent. Then, for any t > 0 , � � � 1 � 2 � √ L τ + ( t ) ( x ) + 1 � law 2 η 2 η ′ = x + 2 t . x 2 x ∈ V x ∈ V Above theorem due to Eisenbaum-Kaspi-Marcus-Rosen-Shi. Similar/related theorems by Ray, Knight, Dynkin, Le Jan, Sznitman, and others. Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 11 / 27
Part II: Application to cover times Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 12 / 27
Gaussian isoperimetric inequality Theorem (Borell and Sudakov-Tsirelson) i ≤ σ 2 for Let η = { η i } i ∈ I be any centered multivariate Gaussian with E η 2 each i. Let X = sup η i . i ∈ I Then, P ( | X − E X | > s · σ ) ≤ 2 (1 − Φ( s )) , where Φ is the Gaussian CDF. In other words, the maximum (or minimum) of a Gaussian process is at least as concentrated as a Gaussian. Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 13 / 27
Fluctuations of the GFF Define x ∈ V E η 2 R = max x , y ∈ V R eff ( x , y ) ≥ max x M = E max v ∈ V η v = − E min v ∈ V η v . Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 14 / 27
Fluctuations of the GFF Define x ∈ V E η 2 R = max x , y ∈ V R eff ( x , y ) ≥ max x M = E max v ∈ V η v = − E min v ∈ V η v . √ Thus, max v ∈ V η v has mean M and fluctuations of order R . Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 14 / 27
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