[PPT] - Exponential concentration of cover times Alex Zhai PowerPoint Presentation

SLIDE 1

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

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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

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Part I: Preliminaries

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 3 / 27

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Our setting

G = (V , E) a simple graph, and fix a starting vertex v0 ∈ V .

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 4 / 27

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Our setting

G = (V , E) a simple graph, and fix a starting vertex v0 ∈ V . We consider continuous time random walks X = {Xt}t∈R+ started at v0:

same as usual simple random walk, except time between jumps is a standard exponential random variable Xt denotes the vertex you’re on at time t

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 4 / 27

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Our setting

G = (V , E) a simple graph, and fix a starting vertex v0 ∈ V . We consider continuous time random walks X = {Xt}t∈R+ started at v0:

same as usual simple random walk, except time between jumps is a standard exponential random variable Xt denotes the vertex you’re on at time t

Define

cover time τcov = the first time all vertices are visited at least once hitting time τhit(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

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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

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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 Reff(x, y) = effective resistance between x and y

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 5 / 27

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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 Reff(x, y) = effective resistance between x and y We can compute Reff(x, y) by solving for a function f : V → R such that ∆f (z) =    1 if z = x −1 if z = y

therwise

Then Reff(x, y) = f (y) − f (x).

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 5 / 27

SLIDE 10

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 Reff(x, y) = effective resistance between x and y We can compute Reff(x, y) by solving for a function f : V → R such that ∆f (z) =    1 if z = x −1 if z = y

therwise

Then Reff(x, y) = f (y) − f (x). Commute time identity: Eτhit(x, y) + Eτhit(y, x) 2 = |E| · Reff(x, y).

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 5 / 27

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Gaussian free field: definition

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 6 / 27

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Gaussian free field: definition

For a graph G = (V , E), the Gaussian free field (GFF) η is a multivariate Gaussian: coordinates ηv indexed by v ∈ V , with ηv0 = 0

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 6 / 27

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Gaussian free field: definition

For a graph G = (V , E), the Gaussian free field (GFF) η is a multivariate Gaussian: coordinates ηv indexed by v ∈ V , with ηv0 = 0 for f ∈ RV with fv0 = 0, [probability of f ] ∝ exp  −1 2

(x,y)∈E

(fx − fy)2  

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 6 / 27

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Gaussian free field: definition

For a graph G = (V , E), the Gaussian free field (GFF) η is a multivariate Gaussian: coordinates ηv indexed by v ∈ V , with ηv0 = 0 for f ∈ RV with fv0 = 0, [probability of f ] ∝ exp  −1 2

(x,y)∈E

(fx − fy)2   equivalently, E (ηx − ηy)2 = Reff(x, y) (note: Eη2

x = Reff(x, v0))

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 6 / 27

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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

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Gaussian free field: example

Let {Bt}t≥0 be a Brownian motion. GFF of a path is η = (0 = B0, B1, . . . , Bn) .

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 8 / 27

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Local times

Reminder: G = (V , E) a graph and Xt a continuous time random walk.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 9 / 27

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Local times

Reminder: G = (V , E) a graph and Xt a continuous time random walk. For x ∈ V and s ∈ R+, define local time Ls(x) = 1 deg(x) s 1 (Xs′ = x) ds′ = 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

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Return times

For any t > 0, define τ +(t) = inf{s ≥ 0 : Ls(v0) ≥ t} = first time that v0 accumulates local time t.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 10 / 27

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Return times

For any t > 0, define τ +(t) = inf{s ≥ 0 : Ls(v0) ≥ t} = first time that v0 accumulates local time t. Remark: τ +

1 deg(v0)

is like the return time of a discrete time

random walk.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 10 / 27

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Return times

For any t > 0, define τ +(t) = inf{s ≥ 0 : Ls(v0) ≥ t} = first time that v0 accumulates local time t. Remark: τ +

1 deg(v0)

is like the return time of a discrete time

random walk. We have Eτ +(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

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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,

Lτ +(t)(x) + 1

2η2

x

x∈V

law

= 1 2

η′

x +

√ 2t 2

x∈V

.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 11 / 27

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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,

Lτ +(t)(x) + 1

2η2

x

x∈V

law

= 1 2

η′

x +

√ 2t 2

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

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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,

Lτ +(t)(x) + 1

2η2

x

x∈V

law

= 1 2

η′

x +

√ 2t 2

x∈V

. Above theorem due to Eisenbaum-Kaspi-Marcus-Rosen-Shi. Similar/related theorems by Ray, Knight, Dynkin, Le Jan, Sznitman, and

thers.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 11 / 27

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Part II: Application to cover times

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 12 / 27

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Gaussian isoperimetric inequality

Theorem (Borell and Sudakov-Tsirelson) Let η = {ηi}i∈I be any centered multivariate Gaussian with Eη2

i ≤ σ2 for

each i. Let X = sup

i∈I

ηi. Then, P (|X − EX| > 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

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Fluctuations of the GFF

Define R = max

x,y∈V Reff(x, y) ≥ max x∈V Eη2 x

M = E max

v∈V ηv = −E min v∈V ηv.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 14 / 27

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Fluctuations of the GFF

Define R = max

x,y∈V Reff(x, y) ≥ max x∈V Eη2 x

M = E max

v∈V ηv = −E min v∈V ηv.

Thus, maxv∈V ηv 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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Fluctuations of the GFF

Define R = max

x,y∈V Reff(x, y) ≥ max x∈V Eη2 x

M = E max

v∈V ηv = −E min v∈V ηv.

Thus, maxv∈V ηv has mean M and fluctuations of order √ R. In many cases, √ R ≪ M.

e.g. complete graph, discrete torus, regular trees doesn’t hold for case of a path

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 14 / 27

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Connection to cover times

Theorem (generalized Ray-Knight)

Lτ +(t)(x) + 1

2η2

x

x∈V

law

= 1 2

η′

x +

√ 2t 2

x∈V

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 15 / 27

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Connection to cover times

Theorem (generalized Ray-Knight)

Lτ +(t)(x) + 1

2η2

x

x∈V

law

= 1 2

η′

x +

√ 2t 2

x∈V

Main observation (Ding-Lee-Peres): τ +(t) < τcov ⇐ ⇒

ne of the Lτ +(t)(x) is 0

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 15 / 27

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Connection to cover times

Theorem (generalized Ray-Knight)

Lτ +(t)(x) + 1

2η2

x

x∈V

law

= 1 2

η′

x +

√ 2t 2

x∈V

Main observation (Ding-Lee-Peres): 2|E| · t ≈ τ +(t) < τcov ⇐ ⇒

ne of the Lτ +(t)(x) is 0

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 15 / 27

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Connection to cover times

Theorem (generalized Ray-Knight)

Lτ +(t)(x) + 1

2η2

x

x∈V

law

= 1 2

η′

x +

√ 2t 2

x∈V

Main observation (Ding-Lee-Peres): 2|E| · t ≈ τ +(t) < τcov ⇐ ⇒

ne of the Lτ +(t)(x) is 0

“ ⇐ ⇒ ”

ne of the
η′

x +

√ 2t

is small

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 15 / 27

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Connection to cover times

Theorem (generalized Ray-Knight)

Lτ +(t)(x) + 1

2η2

x

x∈V

law

= 1 2

η′

x +

√ 2t 2

x∈V

Main observation (Ding-Lee-Peres): 2|E| · t ≈ τ +(t) < τcov ⇐ ⇒

ne of the Lτ +(t)(x) is 0

“ ⇐ ⇒ ”

ne of the
η′

x +

√ 2t

is small

“ ⇐ ⇒ ” E minx∈V η′

x < −

√ 2t

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 15 / 27

SLIDE 35

Connection to cover times

Theorem (generalized Ray-Knight)

Lτ +(t)(x) + 1

2η2

x

x∈V

law

= 1 2

η′

x +

√ 2t 2

x∈V

Main observation (Ding-Lee-Peres): 2|E| · t ≈ τ +(t) < τcov ⇐ ⇒

ne of the Lτ +(t)(x) is 0

“ ⇐ ⇒ ”

ne of the
η′

x +

√ 2t

is small

“ ⇐ ⇒ ” E minx∈V η′

x < −

√ 2t Theorem (Ding-Lee-Peres) Eτcov ≍ |E| ·

−E min

x∈V η′ x

2 = |E| · M2.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 15 / 27

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Statement of the concentration bound

Theorem (Z., following conjecture of Ding) There are universal constants c and C such that P

τcov − |E|M2
≥ |E|(

√ λR · M + λR)

≤ Ce−cλ

for any λ ≥ C.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 16 / 27

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Statement of the concentration bound

Theorem (Z., following conjecture of Ding) There are universal constants c and C such that P

τcov − |E|M2
≥ |E|(

√ λR · M + λR)

≤ Ce−cλ

for any λ ≥ C. Recall: max

x,y∈V Eτhit(x, y) ≍ |E| · R

and Eτcov ≍ |E| · M2.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 16 / 27

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Statement of the concentration bound

Theorem (Z., following conjecture of Ding) There are universal constants c and C such that P

τcov − |E|M2
≥ |E|(

√ λR · M + λR)

≤ Ce−cλ

for any λ ≥ C. Recall: max

x,y∈V Eτhit(x, y) ≍ |E| · R

and Eτcov ≍ |E| · M2. Thus, Eτcov ∼ |E| · M2 whenever max

x,y∈V Eτhit(x, y) ≪ Eτcov.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 16 / 27

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Statement of the concentration bound

Theorem (Z., following conjecture of Ding) There are universal constants c and C such that P

τcov − |E|M2
≥ |E|(

√ λR · M + λR)

≤ Ce−cλ

for any λ ≥ C. Recall: max

x,y∈V Eτhit(x, y) ≍ |E| · R

and Eτcov ≍ |E| · M2. Thus, Eτcov ∼ |E| · M2 whenever max

x,y∈V Eτhit(x, y) ≪ Eτcov.

(Ding proved for trees and bounded degree graphs.)

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 16 / 27

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Upper bound (following Ding-Lee-Peres)

Ax = Lτ +(t)(x) + 1 2η2

x,

Bx = 1 2

η′

x +

√ 2t 2 . Suppose P

Lτ +(t)(x) = 0 for some x
is large.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 17 / 27

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Upper bound (following Ding-Lee-Peres)

Ax = Lτ +(t)(x) + 1 2η2

x,

Bx = 1 2

η′

x +

√ 2t 2 . Suppose P

Lτ +(t)(x) = 0 for some x
is large. Then,

P

min

x∈V Ax < R

≥ P
Lτ +(t)(x) = 0 for some x
· P
η2

x < R

≥0.5

is large,

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 17 / 27

SLIDE 42

Upper bound (following Ding-Lee-Peres)

Ax = Lτ +(t)(x) + 1 2η2

x,

Bx = 1 2

η′

x +

√ 2t 2 . Suppose P

Lτ +(t)(x) = 0 for some x
is large. Then,

P

min

x∈V Ax < R

≥ P
Lτ +(t)(x) = 0 for some x
· P
η2

x < R

≥0.5

is large, so P

min

x∈V Bx < R

= P
min

x∈V Ax < R

is large,

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 17 / 27

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Upper bound (following Ding-Lee-Peres)

Ax = Lτ +(t)(x) + 1 2η2

x,

Bx = 1 2

η′

x +

√ 2t 2 . Suppose P

Lτ +(t)(x) = 0 for some x
is large. Then,

P

min

x∈V Ax < R

≥ P
Lτ +(t)(x) = 0 for some x
· P
η2

x < R

≥0.5

is large, so P

min

x∈V Bx < R

= P
min

x∈V Ax < R

is large, which means

√ 2t can’t be much more than M.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 17 / 27

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Lower bound (following Ding)

Ax = Lτ +(t)(x) + 1 2η2

x,

Bx = 1 2

η′

x +

√ 2t 2 . Suppose √ 2t < M − C √ R.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 18 / 27

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Lower bound (following Ding)

Ax = Lτ +(t)(x) + 1 2η2

x,

Bx = 1 2

η′

x +

√ 2t 2 . Suppose √ 2t < M − C √

R. Then

P

min

x∈V η′ x +

√ 2t < 0

= P
min

x∈V η′ x < −M + C

√ R

is large (for C large, think e.g. C = 10)...

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 18 / 27

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Lower bound (following Ding)

Ax = Lτ +(t)(x) + 1 2η2

x,

Bx = 1 2

η′

x +

√ 2t 2 . Suppose √ 2t < M − C √

R. Then

P

min

x∈V η′ x +

√ 2t < 0

= P
min

x∈V η′ x < −M + C

√ R

is large (for C large, think e.g. C = 10)... and

η′

x +

√ 2t < 0 for some x “ = ⇒ ” Lτ +(t)(x) = 0.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 18 / 27

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Lower bound (following Ding)

Ax = Lτ +(t)(x) + 1 2η2

x,

Bx = 1 2

η′

x +

√ 2t 2 . Suppose √ 2t < M − C √

R. Then

P

min

x∈V η′ x +

√ 2t < 0

= P
min

x∈V η′ x < −M + C

√ R

is large (for C large, think e.g. C = 10)... and

η′

x +

√ 2t < 0 for some x “ = ⇒ ” Lτ +(t)(x) = 0. Important missing step: how to make “ = ⇒ ” rigorous.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 18 / 27

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Concentration of cover times: recap

The transition point of whether Lτ +(t)(x) > 0 for all x ∈ V

ccurs around

√ 2t ≈ M = ⇒ t ≈ 1

2M2.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 19 / 27

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Concentration of cover times: recap

The transition point of whether Lτ +(t)(x) > 0 for all x ∈ V

ccurs around

√ 2t ≈ M = ⇒ t ≈ 1

2M2.

τ +(t) is concentrated around its expectation 2|E| · t as long as R ≪ t, so τcov ≈ τ + 1 2M2

≈ |E| · M2.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 19 / 27

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Concentration of cover times: recap

The transition point of whether Lτ +(t)(x) > 0 for all x ∈ V

ccurs around

√ 2t ≈ M = ⇒ t ≈ 1

2M2.

τ +(t) is concentrated around its expectation 2|E| · t as long as R ≪ t, so τcov ≈ τ + 1 2M2

≈ |E| · M2.

But still need “important missing step”.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 19 / 27

SLIDE 51

Part III: Stochastic domination in the generalized 2nd Ray-Knight theorem

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 20 / 27

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Stochastic domination in the second Ray-Knight theorem

Theorem (variant of theorem of Lupu, conjectured by Ding) We have

Lτ +(t)(x) : x ∈ V
1

√ 2

max
η′

x +

√ 2t, 0

: x ∈ V
,

where denotes stochastic domination.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 21 / 27

SLIDE 53

A graph refinement

Random walk step can be simulated by random walk on refined graph:

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 22 / 27

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A graph refinement

Random walk step can be simulated by random walk on refined graph:

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 22 / 27

SLIDE 55

A graph refinement

Random walk step can be simulated by random walk on refined graph: Refined walk visits x a Geom(n) number of times before going to y or z with equal probability

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 22 / 27

SLIDE 56

A graph refinement

Random walk step can be simulated by random walk on refined graph: Refined walk visits x a Geom(n) number of times before going to y or z with equal probability = ⇒ time spent at x is still exponential

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 22 / 27

SLIDE 57

A graph refinement

The GFFs are also related in a natural way: effective resistances (= GFF covariances) are multiplied by n.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 23 / 27

SLIDE 58

Metric graphs

The limiting object as n → ∞ is known as a metric graph.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 24 / 27

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Metric graphs

The limiting object as n → ∞ is known as a metric graph. In the limit: random walk is a “Brownian motion on edges”. GFF has same law as original graph (up to scaling), with Brownian bridges on edges

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 24 / 27

SLIDE 60

Proof of stochastic domination

Artificially construct a coupling of random walk X and GFFs η and η′ on the metric graph so that

Lτ +(t)(x) + 1

2η2

x

x∈V

= 1 2

η′

x +

√ 2t 2

x∈V

.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 25 / 27

SLIDE 61

Proof of stochastic domination

Artificially construct a coupling of random walk X and GFFs η and η′ on the metric graph so that

Lτ +(t)(x) + 1

2η2

x

x∈V

= 1 2

η′

x +

√ 2t 2

x∈V

. Let U = {set on which Lτ +(t)(x) > 0}. Claim: U is (a.s.) connected.

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 25 / 27

SLIDE 62

Proof of stochastic domination

Artificially construct a coupling of random walk X and GFFs η and η′ on the metric graph so that

Lτ +(t)(x) + 1

2η2

x

x∈V

= 1 2

η′

x +

√ 2t 2

x∈V

. Let U = {set on which Lτ +(t)(x) > 0}. Claim: U is (a.s.) connected. η′

x +

√ 2t = 0 forces Lτ +(t)(x) = 0

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 25 / 27

SLIDE 63

Proof of stochastic domination

Artificially construct a coupling of random walk X and GFFs η and η′ on the metric graph so that

Lτ +(t)(x) + 1

2η2

x

x∈V

= 1 2

η′

x +

√ 2t 2

x∈V

. Let U = {set on which Lτ +(t)(x) > 0}. Claim: U is (a.s.) connected. η′

x +

√ 2t = 0 forces Lτ +(t)(x) = 0 η′

x +

√ 2t can’t change signs on U and is positive at x = v0

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 25 / 27

SLIDE 64

Some open questions

Theorem (generalized Ray-Knight)

Lτ +(t)(x) + 1

2η2

x

x∈V

law

= 1 2

η′

x +

√ 2t 2

x∈V

Only known proofs are by moment calculations. Can we give an explicit coupling? Can be understood relatively well when graph is a path or tree. What about a cycle?

Alex Zhai (azhai@stanford.edu) Exponential concentration of cover times May 17, 2015 26 / 27

SLIDE 65

References

J. Ding. Asymptotics of cover times via Gaussian free fields:

Bounded-degree graphs and general trees. Annals of Probability 42 (2), 464–496 (2014).

J. Ding, J. Lee, and Y. Peres. Cover times, blanket times, and

majorizing measures. Annals of Mathematics 175 (3), 1409–1471 (2012).

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