Stochastic Trace Formula for Closed, Negatively Curved Manifolds

[DRAFT] Last updated by Joe Schaefer on Fri, 12 Apr 2024    source
 

Hyperbolic Honeycomb

My 1997 Ph.D. thesis as a blog entry.

There Is Only One n-Dimensional Wiener Measure μ\mu

Piecewise Linear Approximations to Brownian Motion

The Development Map DM

The Cameron-Martin Formula

Heat Kernels as Radon-Nicodym Derivatives of Weiner Measure

Notation

MM is a negatively curved dim=n\dim=n closed Riemannian manifold with metric gg, metric connection \nabla, and (nonnegative) Laplace-Beltrami Operator ΔM\Delta_M. Let ktΔ/2(x,y)k_{-t\Delta/2}(x,y) represent the heat kernel on MM.

Hence ktΔ/2(x,x)=dDMμ/gdxk_{-t\Delta/2}(x,x) = dDM_*\mu/\sqrt{g}dx is the Radon-Nicodym derivative of n-dimensional Wiener Measure μ\mu, restricted to the pull-back of continuous loop space Ωt(M)x\Omega_t(M)\vert_x, via the inverse of the Weiner measure-preserving development map DMDM. Note: DM1ΩtxDM^{-1}\Omega_t\vert_x is not a loop space.

Ωt0\Omega_t^0 is the space of continuous contractible loops on MM.

Ωt[γ]\Omega_t[\gamma] is the space of continuous loops on MM homotopic to the closed geodesic γ\gamma. Let γ0\gamma_0 be its primitive loop.

DM1Ωt0[γ]DM^{-1}\Omega_t^0[\gamma] is the preimage of continuous contractible loops on MM written as offsets homotopic to γ(s)=DM(s(γ)te1),0st\gamma(s) = DM(\frac{s\ell(\gamma)}{t}\vec{e}^1), 0\leq s \leq t. Think Horocyclic Coordinates — each fiber as the geometric limit of periodic geodesic spheres Sγ0(s)n1(k(γ0)),0st,kS_{\gamma_0(s)}^{n-1}(k\ell(\gamma_0)), 0\leq s \leq t, k\rightarrow\infty, vectorized in the Normal Bundle over γ0\gamma_0. Our curvature constraints imply Horocyclic Coordinates for every γ0\gamma_0 exist as a smooth, DMDM-compatible coordinate map for Ωt0[γ]\Omega_t^0[\gamma].

Now x(τ)+(γ)e1\vec{x}(\tau)+\ell(\gamma)\vec{e}^1 is the undeveloped endpoint of the “offset” kinked geodesic homotopic to γ:DM(x(τ)+s(γ)te1),0st\gamma: DM(\vec{x}(\tau) + \frac{s\ell(\gamma)}{t}\vec{e}^1), 0\leq s \leq t. The curve is periodic with period (γ0)\ell(\gamma_0), and it revisits its kinked starting point DM(x(τ))DM(\vec{x}(\tau)) at time tt, making the computation of its forward derivative J=limstDMDM(x(τ)+(γ)ste1)J=\lim_{s\uparrow t}DM^\prime\vert_{DM(\vec{x}(\tau) + \frac{\ell(\gamma)s}{t}\vec{e}^1)} tractible as a linear automorphism of TDM(x(τ))MT_{DM(\vec{x}(\tau))}M. Importantly, JDM(x(τ)+(γ)e1)J_{DM(\vec{x}(\tau)+\ell(\gamma)\vec{e}^1)} may be constructed using Jacobi Fields, since DMDM is the (iterated) exponential map along any series of connected straight lines in Rn\Reals^n. We will study 1/20t<dX|dX>s 1/2 \int_0^t \bra{dX}\ket{dX}_s, with the solution

Xt=X0+0tJXtdBt\begin{aligned} X_t &= X_0 + \int_0^t \sqrt{J}_{X_t} dB_t \\ \end{aligned}

ZΔ/2(t):=MktΔ/2(x,x)gdx=j=0eλit/2Z_{-\Delta/2}(t) := \int_M k_{-t\Delta/2}(x,x) \sqrt{g}dx = \sum_{j=0}^\infty e^{-\lambda_i t/2} is the trace of the heat kernel.

Finally let us define the following from their Radon-Nicodym derivatives:

DMμ(Ωt):=MDMμ(Ωtxgdx)DMμ(Ωt0):=MDMμ(Ωt0xgdx)DMμ(Ωt[γ]):=MDMμ(Ωt[γ]xgdx)\begin{aligned} DM_*\mu(\Omega_t) &:= \int_M DM_*\mu(\Omega_t\vert_x \sqrt{g}dx)\\ DM_*\mu(\Omega^0_t) &:= \int_M DM_*\mu(\Omega^0_t\vert_x \sqrt{g}dx)\\ DM_*\mu(\Omega_t[\gamma]) &:= \int_M DM_*\mu(\Omega_t[\gamma]\vert_x \sqrt{g}dx) \\ \end{aligned}

Stochastic Trace Formula

ZΔ/2(t)=DMμ(Ωt)=DMμ(Ωt0)+{γ}DMμ(Ωt[γ])DMμ(Ωt0)t0(2πt)n/2(vol(M)+t/6MK(x)gdx+O(t2)) by McKean-SingerDMμ(Ωt[γ])=e(γ)2/2tMDMμ(et<JBBt|Bt>Ωt0[γ]xgdx)  by Cameron-Martin=e(γ)2/2tTγ0ME(etJBΩt0[γ]x(τ))dx1(τ)dxn(τ)dτdDMμ(e(γ)x1(t)Ωt0[γ])dx1(τ)dxn(τ)dτy(τ)t0e<IJDM(x(τ),y(τ))x(τ)|x(τ)>/2t(2πt)(n+1)/2(1+O(t2)) semi-classical limitHorocyclic coordinates:z(τ)x(τ)=x+(γ)e1    M/S1S1kt(x,z)dx=limje(γ)2/2t2πtE(e<JXtjx|x>)=limje(γ)2/2t2πtMj/S1S112πtjndetIJXje(Xj)2/2tXj\begin{aligned} Z_{-\Delta/2}(t) = DM_*\mu(\Omega_t) &= DM_*\mu(\Omega^0_t) + \sum_{\set{\gamma}} DM_*\mu(\Omega_t[\gamma]) \\ DM_*\mu(\Omega_t^0) &\approx_{t\rightarrow 0} (2\pi t)^{-n/2}(vol(M) + t/6\int_M K(x)\sqrt{g} dx + O(t^2))\space \small\text{by McKean-Singer}\\ DM_*\mu(\Omega_t[\gamma]) &= e^{-\ell(\gamma)^2/2t}\int_M DM_*\mu(e^{\bra{J_BB_t}\ket{B_t}} _t \Omega_t^0[\gamma]\vert_x\sqrt{g}dx)\space\small \text{ by Cameron-Martin}\\ &= e^{-\ell(\gamma)^2/2t}\int_{T_{\gamma_0}M} E(e^{J_B}_{t} | \Omega_t^0[\gamma]\vert_{x(\tau)})dx^1(\tau)\dots dx^n(\tau) d\tau\\ \frac{dDM_*\mu(e^{-\ell(\gamma)x^1(t)}\Omega^0_t[\gamma])}{dx^1(\tau)\dots dx^n(\tau)d\tau}\vert_{\vec{y(\tau)}}&\approx_{t\rightarrow 0} \frac{e^{-\bra{|I-J_{DM(\vec{x}(\tau),\vec{y}(\tau))}\vec{x}(\tau)}\ket{\vec{x}(\tau)}/2t}}{(2 \pi t)^{(n+1)/2}}(1+O(t^2))\small \text{ semi-classical limit}\\ \text{Horocyclic coordinates}: z(\tau) - x(\tau) &= x + \ell(\gamma)\vec{e}^1\implies\\ \int_{M/S^1\oplus S^1}k_t(x,z) dx &=\lim_{j\rightarrow\infty}\frac{e^{-\ell(\gamma)^2/2t}}{\sqrt{2\pi t}}E(e^{\bra{J_{X^j_t}\vec{x}}\ket{\vec{x}}})\\ &=\lim_{j\rightarrow\infty}\frac{e^{-\ell(\gamma)^2/2t}}{\sqrt{2\pi t}}\int_{M^j/S^1\oplus S^1}\frac{1}{\sqrt{2\pi t}^{jn}\det|I-J_{X^j}|}e^{-\ell(X^j)^2/2t}X^{j}\\ \end{aligned}

Approximation and the Selberg Trace Formula

In the dim=2\dim = 2 constant curvature κ2-\kappa^2 surface case,

Jx,ydRB=(eκd(x,y)/200eκd(x,y)/2)    <JdRB|JdRB>=eκ(B)dRB12eκ(B)dRB220t<JdB|JdB>=eκ(γ)eκ(γ)detIJγ=(eκ(γ)/2eκ(γ)/2)2\begin{aligned} \sqrt{J_{\vec{x}, \vec{y}}}dRB&= \begin{pmatrix} e^{\kappa d(\vec{x},\vec{y})/2} && 0\\ 0 && e^{-\kappa d(\vec{x},\vec{y})/2}\\ \end{pmatrix} \implies&\\ \bra{\sqrt{J}dRB}\ket{\sqrt{J}dRB} &= e^{\kappa \ell(B)}dRB_1^2 - e^{-\kappa \ell(B)}dRB_2^2\\ \int_0^t \bra{\sqrt{ J}dB}\ket{\sqrt{ J}dB} &= e^{\kappa\ell(\gamma)} - e^{-\kappa\ell(\gamma)}\\ \det I-J_{\gamma} &= (e^{\kappa\ell(\gamma)/2}- e^{-\kappa\ell(\gamma)/2})^2 \end{aligned}

which is constant over (x,τ)(\vec{x},\tau), so the approximation t0\approx_{t\rightarrow 0} line in Equation (2) becomes exact:

DMμ(Ωt[γ])=e(γ)2/2t(γ0)2πt(eκ(γ)/2eκ(γ)/2)γ(t)=γ0(kt)    =ek2(γ0)2/2t(γ0)22πtsinhkκ(γ0)/2\begin{aligned} DM_*\mu(\Omega_t[\gamma]) &= \frac{e^{-\ell(\gamma)^2/2t}\ell(\gamma_0)}{\sqrt{2 \pi t}(e^{\kappa\ell(\gamma)/2} -e^{-\kappa\ell(\gamma)/2})}\\ \gamma(t) = \gamma_0(kt)\implies \\ &=\frac{e^{-k^2\ell(\gamma_0)^2/2t}\ell(\gamma_0)}{2\sqrt{2\pi t}\sinh k\kappa\ell(\gamma_0)/2}\\ \end{aligned}

In the dim=3\dim=3 hyperbolic manifold case, we use complex coordinates (z,zˉ)(z,\bar{z}) on the normal bundle to write

JDM(x+(τ+(γ))e1)=(eκ(γ)000eκ(γ)+iθ(γ)000eκ(γ)iθ(γ))    detIγ0k=1ek(κ(γ0)iθ(γ0))2\begin{aligned} J_{DM(\vec{x}+(\tau+\ell(\gamma))\vec{e}^1)} &= \begin{pmatrix} e^{\kappa\ell(\gamma)} && 0 && 0\\ 0 && e^{-\kappa\ell(\gamma)+i\theta(\gamma)} && 0 \\ 0 && 0 && e^{-\kappa\ell(\gamma)-i\theta(\gamma)} \\ \end{pmatrix}\\ \implies& \\ \det I-{\perp_{\gamma_0}}^k &= |1-e^{-k(\kappa\ell(\gamma_0)-i\theta(\gamma_0))}|^2 \end{aligned}

and since z=x2+ix3    dzˉdz=(dx2idx3)(dx2+idx3)=2idx2dx3z=x^2+ix^3 \implies d\bar{z}\wedge dz= (dx^2-idx^3)\wedge(dx^2+idx^3) = 2idx^2\wedge dx^3, the approximation in Equation (2) again becomes exact:

κ=1    DMμ(Ωt[γ])=ek2(γ0)2/2t(γ0)22πt(1ek(γ0))ek(γ0)/2ek((γ0)/2iθ(γ0))\begin{aligned} \kappa &= 1 \implies \\ DM_*\mu(\Omega_t[\gamma]) &=\frac{e^{-k^2\ell(\gamma_0)^2/2t}\ell(\gamma_0)}{2\sqrt{2\pi t (1-e^{-k\ell(\gamma_0)})}|e^{k\ell(\gamma_0)/2}-e^{-k(\ell(\gamma_0)/2-i\theta(\gamma_0))}|}\\ \end{aligned}